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Jrock4real@Matthew: Thanks a lot! That's a really nice site. I'll definitely start hanging out there. I found this page via a link you provided on another page while looking for help with the \pgfmathsetmacro command; made last night's tasks much less strenuous.
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Matthew Leingang,
Clinical Associate Professor of Mathematics at
New York University@Jrock4real: thanks for the compliments. I've been doing beamer slideshows for my classes for about five years. The Calculus I slides are the ones into which I've put the most work—I'm able to improve them continually from term to term. So your presentations will get gradually better as you accumulate experience and your body of work.
I've been hanging out a lot at http://tex.stackexchange.com/ lately. There are many tikz and beamer users there so feel free to join us!
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Jrock4realHey! I just want to tell you how inspiring your site is. I'm currently a math teacher and I use beamer and tikz for my daily presentations. I thought I was doing too much but, wow, this presentation has 214 slides!! Your students are lucky to have you as a prof. As for me, I'll keep pressing on. I still have a lot to learn about animations in beamer using \pause, \only, \uncover and the like.
1.
Section 5.2
The Definite Integral
V63.0121.002.2010Su, Calculus I
New York University
June 17, 2010
Announcements
. . . . . .
2.
Announcements
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 2 / 32
3.
Objectives
Compute the definite
integral using a limit of
Riemann sums
Estimate the definite
integral using a Riemann
sum (e.g., Midpoint Rule)
Reason with the definite
integral using its
elementary properties.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 3 / 32
4.
Outline
Recall
The definite integral as a limit
Estimating the Definite Integral
Properties of the integral
Comparison Properties of the Integral
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 4 / 32
5.
Cavalieri's method in general
Let f be a positive function defined on the interval [a, b]. We want to
find the area between x = a, x = b, y = 0, and y = f(x).
For each positive integer n, divide up the interval into n pieces. Then
b−a
∆x = . For each i between 1 and n, let xi be the ith step between
n
a and b. So
x0 = a
b−a
x1 = x0 + ∆x = a +
n
b−a
x2 = x1 + ∆x = a + 2 · ...
n
b−a
xi = a + i · ...
n
b−a
. . . . . x
. xn = a + n · =b
. 0 . 1 . . . . i . . .xn−1. n
x x x x n
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 5 / 32
6.
Forming Riemann sums
We have many choices of representative points to approximate the
area in each subinterval.
left endpoints…
∑
n
Ln = f(xi−1 )∆x
i=1
. . . . . . . x
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 6 / 32
7.
Forming Riemann sums
We have many choices of representative points to approximate the
area in each subinterval.
right endpoints…
∑
n
Rn = f(xi )∆x
i=1
. . . . . . . x
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 6 / 32
8.
Forming Riemann sums
We have many choices of representative points to approximate the
area in each subinterval.
midpoints…
∑ ( xi−1 + xi )
n
Mn = f ∆x
2
i=1
. . . . . . . x
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 6 / 32
9.
Forming Riemann sums
We have many choices of representative points to approximate the
area in each subinterval.
the minimum value on the
interval…
. . . . . . . x
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 6 / 32
10.
Forming Riemann sums
We have many choices of representative points to approximate the
area in each subinterval.
the maximum value on the
interval…
. . . . . . . x
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 6 / 32
11.
Forming Riemann sums
We have many choices of representative points to approximate the
area in each subinterval.
…even random points!
. . . . . . . x
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 6 / 32
12.
Forming Riemann sums
We have many choices of representative points to approximate the
area in each subinterval.
…even random points!
. . . . . . . . x
In general, choose ci to be a point in the ith interval [xi−1 , xi ]. Form the
Riemann sum
∑ n
Sn = f(c1 )∆x + f(c2 )∆x + · · · + f(cn )∆x = f(ci )∆x
i=1
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 6 / 32
13.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] .
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
matter what choice of ci we make. . x
.
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
14.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] .
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
matter what choice of ci we make. . x
.
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
15.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 1 = 3.0
L
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
l .
.eft endpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
16.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 2 = 5.25
L
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
l .
.eft endpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
17.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 3 = 6.0
L
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
l .
.eft endpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
18.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 4 = 6.375
L
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
l .
.eft endpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
19.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 5 = 6.59988
L
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
l .
.eft endpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
20.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 6 = 6.75
L
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
l .
.eft endpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
21.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 7 = 6.85692
L
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
l .
.eft endpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
22.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 8 = 6.9375
L
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
l .
.eft endpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
23.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 9 = 6.99985
L
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
l .
.eft endpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
24.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 10 = 7.04958
L
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
l .
.eft endpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
25.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 11 = 7.09064
L
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
l .
.eft endpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
26.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 12 = 7.125
L
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
l .
.eft endpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
27.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 13 = 7.15332
L
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
l .
.eft endpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
28.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 14 = 7.17819
L
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
l .
.eft endpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
29.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 15 = 7.19977
L
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
l .
.eft endpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
30.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 16 = 7.21875
L
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
l .
.eft endpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
31.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 17 = 7.23508
L
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
l .
.eft endpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
32.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 18 = 7.24927
L
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
l .
.eft endpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
33.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 19 = 7.26228
L
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
l .
.eft endpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
34.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 20 = 7.27443
L
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
l .
.eft endpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
35.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 21 = 7.28532
L
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
l .
.eft endpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
36.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 22 = 7.29448
L
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
l .
.eft endpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
37.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 23 = 7.30406
L
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
l .
.eft endpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
38.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 24 = 7.3125
L
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
l .
.eft endpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
39.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 25 = 7.31944
L
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
l .
.eft endpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
40.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 26 = 7.32559
L
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
l .
.eft endpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
41.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 27 = 7.33199
L
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
l .
.eft endpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
42.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 28 = 7.33798
L
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
l .
.eft endpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
43.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 29 = 7.34372
L
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
l .
.eft endpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
44.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 30 = 7.34882
L
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
l .
.eft endpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
45.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 1 = 12.0
R
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
r .
. ight endpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
46.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 2 = 9.75
R
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
r .
. ight endpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
47.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 3 = 9.0
R
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
r .
. ight endpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
48.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 4 = 8.625
R
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
r .
. ight endpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
49.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 5 = 8.39969
R
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
r .
. ight endpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
50.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 6 = 8.25
R
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
r .
. ight endpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
51.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 7 = 8.14236
R
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
r .
. ight endpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
52.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 8 = 8.0625
R
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
r .
. ight endpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
53.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 9 = 7.99974
R
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
r .
. ight endpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
54.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 10 = 7.94933
R
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
r .
. ight endpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
55.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 11 = 7.90868
R
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
r .
. ight endpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
56.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 12 = 7.875
R
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
r .
. ight endpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
57.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 13 = 7.84541
R
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
r .
. ight endpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
58.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 14 = 7.8209
R
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
r .
. ight endpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
59.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 15 = 7.7997
R
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
r .
. ight endpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
60.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 16 = 7.78125
R
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
r .
. ight endpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
61.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 17 = 7.76443
R
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
r .
. ight endpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
62.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 18 = 7.74907
R
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
r .
. ight endpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
63.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 19 = 7.73572
R
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
r .
. ight endpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
64.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 20 = 7.7243
R
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
r .
. ight endpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
65.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 21 = 7.7138
R
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
r .
. ight endpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
66.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 22 = 7.70335
R
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
r .
. ight endpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
67.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 23 = 7.69531
R
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
r .
. ight endpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
68.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 24 = 7.6875
R
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
r .
. ight endpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
69.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 25 = 7.67934
R
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
r .
. ight endpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
70.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 26 = 7.6715
R
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
r .
. ight endpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
71.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 27 = 7.66508
R
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
r .
. ight endpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
72.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 28 = 7.6592
R
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
r .
. ight endpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
73.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 29 = 7.65388
R
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
r .
. ight endpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
74.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 30 = 7.64864
R
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
r .
. ight endpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
75.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 1 = 7.5
M
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. idpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
76.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 2 = 7.5
M
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. idpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
77.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 3 = 7.5
M
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. idpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
78.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 4 = 7.5
M
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. idpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
79.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 5 = 7.4998
M
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. idpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
80.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 6 = 7.5
M
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. idpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
81.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 7 = 7.4996
M
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. idpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
82.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 8 = 7.5
M
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. idpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
83.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 9 = 7.49977
M
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. idpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
84.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 10 = 7.49947
M
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. idpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
85.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 11 = 7.49966
M
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. idpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
86.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 12 = 7.5
M
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. idpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
87.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 13 = 7.49937
M
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. idpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
88.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 14 = 7.49954
M
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. idpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
89.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 15 = 7.49968
M
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. idpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
90.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 16 = 7.49988
M
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. idpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
91.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 17 = 7.49974
M
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. idpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
92.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 18 = 7.49916
M
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. idpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
93.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 19 = 7.49898
M
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. idpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
94.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 20 = 7.4994
M
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. idpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
95.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 21 = 7.49951
M
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. idpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
96.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 22 = 7.49889
M
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. idpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
97.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 23 = 7.49962
M
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. idpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
98.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 24 = 7.5
M
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. idpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
99.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 25 = 7.49939
M
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. idpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
100.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 26 = 7.49847
M
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. idpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
101.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 27 = 7.4985
M
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. idpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
102.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 28 = 7.4986
M
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. idpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
103.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 29 = 7.49878
M
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. idpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
104.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 30 = 7.49872
M
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. idpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
105.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 1 = 12.0
U
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. aximum points
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
106.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 2 = 10.55685
U
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. aximum points
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
107.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 3 = 10.0379
U
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. aximum points
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
108.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 4 = 9.41515
U
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. aximum points
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
109.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 5 = 8.96004
U
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. aximum points
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
110.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 6 = 8.76895
U
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. aximum points
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
111.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 7 = 8.6033
U
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. aximum points
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
112.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 8 = 8.45757
U
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. aximum points
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
113.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 9 = 8.34564
U
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. aximum points
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
114.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 10 = 8.27084
U
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. aximum points
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
115.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 11 = 8.20132
U
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. aximum points
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
116.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 12 = 8.13838
U
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. aximum points
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
117.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 13 = 8.0916
U
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. aximum points
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
118.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 14 = 8.05139
U
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. aximum points
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
119.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 15 = 8.01364
U
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. aximum points
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
120.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 16 = 7.98056
U
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. aximum points
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
121.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 17 = 7.9539
U
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. aximum points
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
122.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 18 = 7.92815
U
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. aximum points
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
123.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 19 = 7.90414
U
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. aximum points
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
124.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 20 = 7.88504
U
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. aximum points
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
125.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 21 = 7.86737
U
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. aximum points
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
126.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 22 = 7.84958
U
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. aximum points
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
127.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 23 = 7.83463
U
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. aximum points
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
128.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 24 = 7.82187
U
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. aximum points
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
129.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 25 = 7.80824
U
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. aximum points
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
130.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 26 = 7.79504
U
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. aximum points
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
131.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 27 = 7.78429
U
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. aximum points
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
132.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 28 = 7.77443
U
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. aximum points
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
133.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 29 = 7.76495
U
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. aximum points
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
134.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 30 = 7.7558
U
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. aximum points
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
135.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 1 = 3.0
L
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. inimum points
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
136.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 2 = 4.44312
L
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. inimum points
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
137.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 3 = 4.96208
L
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. inimum points
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
138.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 4 = 5.58484
L
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. inimum points
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
139.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 5 = 6.0395
L
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. inimum points
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
140.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 6 = 6.23103
L
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. inimum points
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
141.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 7 = 6.39577
L
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. inimum points
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
142.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 8 = 6.54242
L
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. inimum points
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
143.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 9 = 6.65381
L
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. inimum points
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
144.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 10 = 6.72797
L
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. inimum points
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
145.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 11 = 6.7979
L
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. inimum points
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
146.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 12 = 6.8616
L
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. inimum points
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
147.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 13 = 6.90704
L
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. inimum points
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
148.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 14 = 6.94762
L
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. inimum points
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
149.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 15 = 6.98575
L
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. inimum points
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
150.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 16 = 7.01942
L
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. inimum points
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
151.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 17 = 7.04536
L
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. inimum points
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
152.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 18 = 7.07005
L
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. inimum points
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
153.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 19 = 7.09364
L
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. inimum points
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
154.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 20 = 7.1136
L
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. inimum points
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
155.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 21 = 7.13155
L
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. inimum points
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
156.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 22 = 7.14804
L
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. inimum points
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
157.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 23 = 7.16441
L
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. inimum points
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
158.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 24 = 7.17812
L
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. inimum points
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
159.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 25 = 7.19025
L
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. inimum points
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
160.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 26 = 7.2019
L
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. inimum points
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
161.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 27 = 7.21265
L
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. inimum points
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
162.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 28 = 7.22269
L
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. inimum points
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
163.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 29 = 7.23251
L
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. inimum points
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
164.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 30 = 7.24162
L
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. inimum points
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 7 / 32
165.
Outline
Recall
The definite integral as a limit
Estimating the Definite Integral
Properties of the integral
Comparison Properties of the Integral
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 8 / 32
166.
The definite integral as a limit
Definition
If f is a function defined on [a, b], the definite integral of f from a to b
is the number ∫ b ∑n
f(x) dx = lim f(ci ) ∆x
a ∆x→0
i=1
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 9 / 32
167.
Notation/Terminology
∫ b ∑
n
f(x) dx = lim f(ci ) ∆x
a ∆x→0
i=1
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 10 / 32
168.
Notation/Terminology
∫ b ∑
n
f(x) dx = lim f(ci ) ∆x
a ∆x→0
i=1
∫
— integral sign (swoopy S)
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 10 / 32
169.
Notation/Terminology
∫ b ∑
n
f(x) dx = lim f(ci ) ∆x
a ∆x→0
i=1
∫
— integral sign (swoopy S)
f(x) — integrand
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 10 / 32
170.
Notation/Terminology
∫ b ∑
n
f(x) dx = lim f(ci ) ∆x
a ∆x→0
i=1
∫
— integral sign (swoopy S)
f(x) — integrand
a and b — limits of integration (a is the lower limit and b the
upper limit)
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 10 / 32
171.
Notation/Terminology
∫ b ∑
n
f(x) dx = lim f(ci ) ∆x
a ∆x→0
i=1
∫
— integral sign (swoopy S)
f(x) — integrand
a and b — limits of integration (a is the lower limit and b the
upper limit)
dx — ??? (a parenthesis? an infinitesimal? a variable?)
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 10 / 32
172.
Notation/Terminology
∫ b ∑
n
f(x) dx = lim f(ci ) ∆x
a ∆x→0
i=1
∫
— integral sign (swoopy S)
f(x) — integrand
a and b — limits of integration (a is the lower limit and b the
upper limit)
dx — ??? (a parenthesis? an infinitesimal? a variable?)
The process of computing an integral is called integration or
quadrature
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 10 / 32
173.
The limit can be simplified
Theorem
If f is continuous on [a, b] or if f has only finitely many jump
discontinuities, then f is integrable on [a, b]; that is, the definite integral
∫ b
f(x) dx exists.
a
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 11 / 32
174.
The limit can be simplified
Theorem
If f is continuous on [a, b] or if f has only finitely many jump
discontinuities, then f is integrable on [a, b]; that is, the definite integral
∫ b
f(x) dx exists.
a
Theorem
If f is integrable on [a, b] then
∫ b ∑
n
f(x) dx = lim f(xi )∆x,
a n→∞
i=1
where
b−a
∆x = and xi = a + i ∆x
n
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 11 / 32
175.
Example: Integral of x
Example
∫ 3
Find x dx
0
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 12 / 32
176.
Example: Integral of x
Example
∫ 3
Find x dx
0
Solution
3 3i
For any n we have ∆x = and xi = . So
n n
∑
n ∑ ( 3i ) ( 3 )
n
9 ∑
n
Rn = f(xi ) ∆x = = 2 i
n n n
i=1 i=1 i=1
9 n(n + 1) 9
= 2· −→
n 2 2
∫ 3
9
So x dx = = 4.5
0 2
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 12 / 32
177.
Example: Integral of x2
Example
∫ 3
Find x2 dx
0
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 13 / 32
178.
Example: Integral of x2
Example
∫ 3
Find x2 dx
0
Solution
3 3i
For any n we have ∆x = and xi = . So
n n
∑
n ∑ ( 3i )2 ( 3 ) 27 ∑
n n
Rn = f(xi ) ∆x = = 3 i2
n n n
i=1 i=1 i=1
27 n(n + 1)(2n + 1) 27
= 3 · −→ =9
n 6 3
∫ 3
So x2 dx = 9
0
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 13 / 32
179.
Example: Integral of x3
Example
∫ 3
Find x3 dx
0
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 14 / 32
180.
Example: Integral of x3
Example
∫ 3
Find x3 dx
0
Solution
3 3i
For any n we have ∆x = and xi = . So
n n
∑
n ∑ ( 3i )3 ( 3 ) 81 ∑
n n
Rn = f(xi ) ∆x = = 4 i3
n n n
i=1 i=1 i=1
81 n2 (n + 1)2 81
= 4 · −→
n 4 4
∫ 3
81
So x3 dx = = 20.25
0 4
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 14 / 32
181.
Outline
Recall
The definite integral as a limit
Estimating the Definite Integral
Properties of the integral
Comparison Properties of the Integral
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 15 / 32
182.
Estimating the Definite Integral
Example
∫ 1
4
Estimate dx using M4 .
0 1 + x2
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 16 / 32
183.
Estimating the Definite Integral
Example
∫ 1
4
Estimate dx using M4 .
0 1 + x2
Solution
1 1 3
We have x0 = 0, x1 = , x2 = , x3 = , x4 = 1.
4 2 4
1 3 5 7
So c1 = , c2 = , c3 = , c4 = .
8 8 8 8
( )
1 4 4 4 4
M4 = + + +
4 1 + (1/8)2 1 + (3/8)2 1 + (5/8)2 1 + (7/8)2
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 16 / 32
186.
Estimating the Definite Integral (Continued)
Example
∫ 1
4
Estimate dx using L4 and R4
0 1 + x2
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 17 / 32
188.
Outline
Recall
The definite integral as a limit
Estimating the Definite Integral
Properties of the integral
Comparison Properties of the Integral
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 18 / 32
189.
Properties of the integral
Theorem (Additive Properties of the Integral)
Let f and g be integrable functions on [a, b] and c a constant. Then
∫ b
1. c dx = c(b − a)
a
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 19 / 32
190.
Properties of the integral
Theorem (Additive Properties of the Integral)
Let f and g be integrable functions on [a, b] and c a constant. Then
∫ b
1. c dx = c(b − a)
a
∫ b ∫ b ∫ b
2. [f(x) + g(x)] dx = f(x) dx + g(x) dx.
a a a
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 19 / 32
191.
Properties of the integral
Theorem (Additive Properties of the Integral)
Let f and g be integrable functions on [a, b] and c a constant. Then
∫ b
1. c dx = c(b − a)
a
∫ b ∫ b ∫ b
2. [f(x) + g(x)] dx = f(x) dx + g(x) dx.
a a a
∫ b ∫ b
3. cf(x) dx = c f(x) dx.
a a
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 19 / 32
192.
Properties of the integral
Theorem (Additive Properties of the Integral)
Let f and g be integrable functions on [a, b] and c a constant. Then
∫ b
1. c dx = c(b − a)
a
∫ b ∫ b ∫ b
2. [f(x) + g(x)] dx = f(x) dx + g(x) dx.
a a a
∫ b ∫ b
3. cf(x) dx = c f(x) dx.
a a
∫ b ∫ b ∫ b
4. [f(x) − g(x)] dx = f(x) dx − g(x) dx.
a a a
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 19 / 32
193.
Proofs
Proofs.
When integrating a constant function c, each Riemann sum
equals c(b − a).
A Riemann sum for f + g equals a Riemann sum for f plus a
Riemann sum for g. Using the sum rule for limits, the integral of a
sum is the sum of the integrals.
Ditto for constant multiples
Ditto for differences
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 20 / 32
194.
Example
∫ 3( )
Find x3 − 4.5x2 + 5.5x + 1 dx
0
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 21 / 32
196.
Example
∫ 3( )
Find x3 − 4.5x2 + 5.5x + 1 dx
0
Solution
∫ 3
(x3 −4.5x2 + 5.5x + 1) dx
0
∫ 3 ∫ 3 ∫ 3 ∫ 3
= x dx − 4.5
3 2
x dx + 5.5 x dx + 1 dx
0 0 0 0
= 20.25 − 4.5 · 9 + 5.5 · 4.5 + 3 · 1 = 7.5
(This is the function we were estimating the integral of before)
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 21 / 32
197.
Theorem of the (previous) Day
Theorem
If f is a continuous function on [a, b] . . 15 = 7.49968
M
or has finitely many jump
discontinuities, then
{ n }
∑
lim Sn = lim f(ci )∆x
n→∞ n→∞
i=1
exists and is the same value no
. x
.
matter what choice of ci we make.
m .
. idpoints
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 22 / 32
198.
More Properties of the Integral
Conventions: ∫ ∫
a b
f(x) dx = − f(x) dx
b a
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 23 / 32
199.
More Properties of the Integral
Conventions: ∫ ∫
a b
f(x) dx = − f(x) dx
b a
∫ a
f(x) dx = 0
a
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 23 / 32
200.
More Properties of the Integral
Conventions: ∫ ∫
a b
f(x) dx = − f(x) dx
b a
∫ a
f(x) dx = 0
a
This allows us to have
∫ c ∫ b ∫ c
5. f(x) dx = f(x) dx + f(x) dx for all a, b, and c.
a a b
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 23 / 32
201.
Example
Suppose f and g are functions with
∫ 4
f(x) dx = 4
0
∫ 5
f(x) dx = 7
0
∫ 5
g(x) dx = 3.
0
Find
∫ 5
(a) [2f(x) − g(x)] dx
0
∫ 5
(b) f(x) dx.
4
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 24 / 32
202.
Solution
We have
(a)
∫ 5 ∫ 5 ∫ 5
[2f(x) − g(x)] dx = 2 f(x) dx − g(x) dx
0 0 0
= 2 · 7 − 3 = 11
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 25 / 32
204.
Outline
Recall
The definite integral as a limit
Estimating the Definite Integral
Properties of the integral
Comparison Properties of the Integral
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 26 / 32
205.
Comparison Properties of the Integral
Theorem
Let f and g be integrable functions on [a, b].
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 27 / 32
206.
Comparison Properties of the Integral
Theorem
Let f and g be integrable functions on [a, b].
6. If f(x) ≥ 0 for all x in [a, b], then
∫ b
f(x) dx ≥ 0
a
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 27 / 32
207.
Comparison Properties of the Integral
Theorem
Let f and g be integrable functions on [a, b].
6. If f(x) ≥ 0 for all x in [a, b], then
∫ b
f(x) dx ≥ 0
a
7. If f(x) ≥ g(x) for all x in [a, b], then
∫ b ∫ b
f(x) dx ≥ g(x) dx
a a
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 27 / 32
208.
Comparison Properties of the Integral
Theorem
Let f and g be integrable functions on [a, b].
6. If f(x) ≥ 0 for all x in [a, b], then
∫ b
f(x) dx ≥ 0
a
7. If f(x) ≥ g(x) for all x in [a, b], then
∫ b ∫ b
f(x) dx ≥ g(x) dx
a a
8. If m ≤ f(x) ≤ M for all x in [a, b], then
∫ b
m(b − a) ≤ f(x) dx ≤ M(b − a)
a
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 27 / 32
209.
The integral of a nonnegative function is nonnegative
Proof.
If f(x) ≥ 0 for all x in [a, b], then for any number of divisions n and
choice of sample points {ci }:
∑
n ∑
n
Sn = f(ci ) ∆x ≥ 0 · ∆x = 0
i=1 i=1
≥0
Since Sn ≥ 0 for all n, the limit of {Sn } is nonnegative, too:
∫ b
f(x) dx = lim Sn ≥ 0
a n→∞
≥0
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 28 / 32
210.
The definite integral is “increasing"
Proof.
Let h(x) = f(x) − g(x). If f(x) ≥ g(x) for all x in [a, b], then h(x) ≥ 0 for
all x in [a, b]. So by the previous property
∫ b
h(x) dx ≥ 0
a
This means that
∫ b ∫ b ∫ b ∫ b
f(x) dx − g(x) dx = (f(x) − g(x)) dx = h(x) dx ≥ 0
a a a a
So ∫ ∫
b b
f(x) dx ≥ g(x) dx
a a
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 29 / 32
211.
Bounding the integral using bounds of the function
Proof.
If m ≤ f(x) ≤ M on for all x in [a, b], then by the previous property
∫ b ∫ b ∫ b
m dx ≤ f(x) dx ≤ M dx
a a a
By Property ??, the integral of a constant function is the product of the
constant and the width of the interval. So:
∫ b
m(b − a) ≤ f(x) dx ≤ M(b − a)
a
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 30 / 32
212.
Example
∫ 2
1
Estimate dx using the comparison properties.
1 x
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 31 / 32
213.
Example
∫ 2
1
Estimate dx using the comparison properties.
1 x
Solution
Since
1 1
≤x≤
2 1
for all x in [1, 2], we have
∫ 2
1 1
·1≤ dx ≤ 1 · 1
2 1 x
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 31 / 32
214.
Summary
The definite integral is a limit of Riemann Sums
The definite integral can be estimated with Riemann Sums
The definite integral can be distributed across sums and constant
multiples of functions
The definite integral can be bounded using bounds for the function
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.2 The Definite Integral June 17, 2010 32 / 32
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