Transcript of "Lesson 24: The Definite Integral (Section 10 version)"
1.
Section 5.2
The Deﬁnite Integral
V63.0121, Calculus I
April 15, 2009
Announcements
My ofﬁce is now WWH 624
Final Exam Friday, May 8, 2:00–3:50pm
. . . . . .
2.
Outline
Recall
The deﬁnite integral as a limit
Estimating the Deﬁnite Integral
Properties of the integral
Comparison Properties of the Integral
. . . . . .
3.
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.
b−a
. For each i between 1 and n, let xi be the ith
Then ∆x =
n
step between 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
xn = a + n · =b
n
.. . .. x
.
. 0 . 1 . . . . i . . .. n−1. n
x x. x. x x
. . . . . .
4.
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
.
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
. . . . . .
5.
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
.
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
. . . . . .
6.
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
.
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
. . . . . .
7.
Forming Riemann sums
We have many choices of representative points to approximate
the area in each subinterval.
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
. . . . . .
8.
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
. x
.
matter what choice of ci we
made.
. . . . . .
9.
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
. x
.
matter what choice of ci we
made.
. . . . . .
10.
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
. . . x
.
matter what choice of ci we
made.
. . . . . .
11.
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
. . . . x
.
matter what choice of ci we
made.
. . . . . .
12.
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
. . . . . x
.
matter what choice of ci we
made.
. . . . . .
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
. . . . . . x
.
matter what choice of ci we
made.
. . . . . .
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
....... x
.
matter what choice of ci we
made.
. . . . . .
15.
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
........ x
.
matter what choice of ci we
made.
. . . . . .
16.
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
......... x
.
matter what choice of ci we
made.
. . . . . .
17.
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
.......... x
.
matter what choice of ci we
made.
. . . . . .
18.
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
........... x
.
matter what choice of ci we
made.
. . . . . .
19.
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
............ x
.
matter what choice of ci we
made.
. . . . . .
20.
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
............. x
.
matter what choice of ci we
made.
. . . . . .
21.
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
.............. x
.
matter what choice of ci we
made.
. . . . . .
22.
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
............... x
.
matter what choice of ci we
made.
. . . . . .
23.
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
................ x
.
matter what choice of ci we
made.
. . . . . .
24.
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
................. x
.
matter what choice of ci we
made.
. . . . . .
25.
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
.................. x
.
matter what choice of ci we
made.
. . . . . .
26.
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
................... x
.
matter what choice of ci we
made.
. . . . . .
27.
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
.................... x
.
matter what choice of ci we
made.
. . . . . .
28.
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
.....................
. x
.
matter what choice of ci we
made.
. . . . . .
29.
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
......................
. x
.
matter what choice of ci we
made.
. . . . . .
30.
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
.......................
. x
.
matter what choice of ci we
made.
. . . . . .
31.
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
........................
. x
.
matter what choice of ci we
made.
. . . . . .
32.
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
.........................
. x
.
matter what choice of ci we
made.
. . . . . .
33.
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
..........................
. x
.
matter what choice of ci we
made.
. . . . . .
34.
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
...........................
. x
.
matter what choice of ci we
made.
. . . . . .
35.
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
............................
. x
.
matter what choice of ci we
made.
. . . . . .
36.
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
.............................
. x
.
matter what choice of ci we
made.
. . . . . .
37.
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
..............................
. x
.
matter what choice of ci we
made.
. . . . . .
38.
Outline
Recall
The deﬁnite integral as a limit
Estimating the Deﬁnite Integral
Properties of the integral
Comparison Properties of the Integral
. . . . . .
39.
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
∆x→0
a i=1
. . . . . .
40.
Notation/Terminology
∫ b
f(x) dx
a
. . . . . .
41.
Notation/Terminology
∫ b
f(x) dx
a
∫
— integral sign (swoopy S)
. . . . . .
42.
Notation/Terminology
∫ b
f(x) dx
a
∫
— integral sign (swoopy S)
f(x) — integrand
. . . . . .
43.
Notation/Terminology
∫ b
f(x) dx
a
∫
— integral sign (swoopy S)
f(x) — integrand
a and b — limits of integration (a is the lower limit and b
the upper limit)
. . . . . .
44.
Notation/Terminology
∫ b
f(x) dx
a
∫
— 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?)
. . . . . .
45.
Notation/Terminology
∫ b
f(x) dx
a
∫
— 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
. . . . . .
46.
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
∫b
integral f(x) dx exists.
a
. . . . . .
47.
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
∫b
integral f(x) dx exists.
a
Theorem
If f is integrable on [a, b] then
∫ n
∑
b
f(x) dx = lim f(xi )∆x,
n→∞
a i=1
where
b−a
and xi = a + i ∆x
∆x =
n
. . . . . .
48.
Outline
Recall
The deﬁnite integral as a limit
Estimating the Deﬁnite Integral
Properties of the integral
Comparison Properties of the Integral
. . . . . .
49.
Estimating the Deﬁnite Integral
Given a partition of [a, b] into n pieces, let ¯i be the midpoint of
x
[xi−1 , xi ]. Deﬁne
n
∑
Mn = f(¯i ) ∆x.
x
i=1
. . . . . .
50.
Example
∫ 1
4
dx using the midpoint rule and four divisions.
Estimate
1 + x2
0
. . . . . .
51.
Example
∫ 1
4
dx using the midpoint rule and four divisions.
Estimate
1 + x2
0
Solution
1 1 3
The partition is 0 < < < < 1, so the estimate is
4 2 4
( )
1 4 4 4 4
M4 = + + +
2 2 2 1 + (7/8)2
4 1 + (1/8) 1 + (3/8) 1 + (5/8)
. . . . . .
52.
Example
∫ 1
4
dx using the midpoint rule and four divisions.
Estimate
1 + x2
0
Solution
1 1 3
The partition is 0 < < < < 1, so the estimate is
4 2 4
( )
1 4 4 4 4
M4 = + + +
4 1 + (1/8)2 1 + (3/8)2 1 + (5/8)2 1 + (7/8)2
( )
1 4 4 4 4
= + + +
4 65/64 73/64 89/64 113/64
. . . . . .
53.
Example
∫ 1
4
dx using the midpoint rule and four divisions.
Estimate
1 + x2
0
Solution
1 1 3
The partition is 0 < < < < 1, so the estimate is
4 2 4
( )
1 4 4 4 4
M4 = + + +
4 1 + (1/8)2 1 + (3/8)2 1 + (5/8)2 1 + (7/8)2
( )
1 4 4 4 4
= + + +
4 65/64 73/64 89/64 113/64
150, 166, 784
≈ 3.1468
=
47, 720, 465
. . . . . .
54.
Outline
Recall
The deﬁnite integral as a limit
Estimating the Deﬁnite Integral
Properties of the integral
Comparison Properties of the Integral
. . . . . .
55.
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
c dx = c(b − a)
1.
a
. . . . . .
56.
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
c dx = c(b − a)
1.
a
∫ ∫ ∫
b b b
[f(x) + g(x)] dx = f(x) dx + g(x) dx.
2.
a a a
. . . . . .
57.
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
c dx = c(b − a)
1.
a
∫ ∫ ∫
b b b
[f(x) + g(x)] dx = f(x) dx + g(x) dx.
2.
a a a
∫ ∫
b b
cf(x) dx = c f(x) dx.
3.
a a
. . . . . .
58.
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
c dx = c(b − a)
1.
a
∫ ∫ ∫
b b b
[f(x) + g(x)] dx = f(x) dx + g(x) dx.
2.
a a a
∫ ∫
b b
cf(x) dx = c f(x) dx.
3.
a a
∫ ∫ ∫
b b b
[f(x) − g(x)] dx = f(x) dx − g(x) dx.
4.
a a a
. . . . . .
59.
More Properties of the Integral
Conventions: ∫ ∫
a b
f(x) dx = − f(x) dx
b a
. . . . . .
60.
More Properties of the Integral
Conventions: ∫ ∫
a b
f(x) dx = − f(x) dx
b a
∫ a
f(x) dx = 0
a
. . . . . .
61.
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
f(x) dx = f(x) dx + f(x) dx for all a, b, and c.
5.
a a b
. . . . . .
62.
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
[2f(x) − g(x)] dx
(a)
0
∫5
f(x) dx.
(b)
4
. . . . . .
65.
Outline
Recall
The deﬁnite integral as a limit
Estimating the Deﬁnite Integral
Properties of the integral
Comparison Properties of the Integral
. . . . . .
66.
Comparison Properties of the Integral
Theorem
Let f and g be integrable functions on [a, b].
. . . . . .
67.
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
. . . . . .
68.
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
. . . . . .
69.
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
. . . . . .
70.
Example
∫ 2
1
dx using the comparison properties.
Estimate
x
1
. . . . . .
71.
Example
∫ 2
1
dx using the comparison properties.
Estimate
x
1
Solution
Since
1 1
≤x≤
2 1
for all x in [1, 2], we have
∫ 2
1 1
·1≤ dx ≤ 1 · 1
x
2 1
. . . . . .
A particular slide catching your eye?
Clipping is a handy way to collect important slides you want to go back to later.
Be the first to comment