This document provides an overview of beginning calculus concepts related to antiderivatives, definite integrals, and computing areas under curves. It discusses evaluating antiderivatives using substitution and guessing methods. It also explains how to compute definite integrals using Riemann sums and taking the limit of partition widths approaching zero. Examples are provided to illustrate computing areas under curves for basic functions like polynomials and finding net areas when functions are sometimes positive and sometimes negative.

1. Beginning Calculus
- Antiderivatives and The De…nite Integrals -
Shahrizal Shamsuddin Norashiqin Mohd Idrus
Department of Mathematics,
FSMT - UPSI
(LECTURE SLIDES SERIES)
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2. Anti Derivatives Area Under a Curve Riemann Sums Net Area Properties of the De…nite Integral
Learning Outcomes
Use substitution and advanced guessing methods to evaluate anti
derivatives.
Compute Riemann Sums.
Compute areas under the curve and net areas.
State and apply properties of the de…nite integrals.
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3. Anti Derivatives Area Under a Curve Riemann Sums Net Area Properties of the De…nite Integral
Anti Derivatives
G (x) =
Z
g (x) dx
G (x) is called the anti derivative of g, or the inde…nite integral
of g.
G0 (x) = g (x)
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4. Anti Derivatives Area Under a Curve Riemann Sums Net Area Properties of the De…nite Integral
Example
R
sin xdx = cos x + C
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5. Anti Derivatives Area Under a Curve Riemann Sums Net Area Properties of the De…nite Integral
Example
R
xadx =
xa+1
a + 1
+ C, for a 6= 1
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6. Anti Derivatives Area Under a Curve Riemann Sums Net Area Properties of the De…nite Integral
Example
R dx
x
= ln jxj + C
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7. Anti Derivatives Area Under a Curve Riemann Sums Net Area Properties of the De…nite Integral
More Examples
R
sec2 xdx = tan x + C
R dx
p
1 x2
= sin 1 x + C
R dx
1 + x2
= tan 1 x + C
R
ex dx = ex + C
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8. Anti Derivatives Area Under a Curve Riemann Sums Net Area Properties of the De…nite Integral
Uniqueness of anti derivatives up to a constant
Theorem 1
If F0 = G0, then F (x) = G (x) + C.
Proof.
Suppose F0 = G0. Then,
(F G)0
= F0
G0
= 0
F (x) G (x) = C
) F (x) = G (x) + C
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9. Anti Derivatives Area Under a Curve Riemann Sums Net Area Properties of the De…nite Integral
Method of Substitution - For Di¤erential Notation
R
x3 x4 + 2
5
dx
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10. Anti Derivatives Area Under a Curve Riemann Sums Net Area Properties of the De…nite Integral
Example
R xdx
p
1 + x2
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11. Anti Derivatives Area Under a Curve Riemann Sums Net Area Properties of the De…nite Integral
Example
R
e6x dx
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12. Anti Derivatives Area Under a Curve Riemann Sums Net Area Properties of the De…nite Integral
Example - Advanced Guessing
R
xe x2
dx
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13. Anti Derivatives Area Under a Curve Riemann Sums Net Area Properties of the De…nite Integral
Example
R dx
x ln x
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14. Anti Derivatives Area Under a Curve Riemann Sums Net Area Properties of the De…nite Integral
Area Under a Curve
b
( )dxxf
b
a∫
a
( )xfy =
Area under a curve =
R b
a f (x) dx
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15. Anti Derivatives Area Under a Curve Riemann Sums Net Area Properties of the De…nite Integral
Area Under a Curve
To compute the area under a curve:
b
L
a
1 Divide into n rectangles
2 Add up the areas
3 Take the limit as n ! ∞ (the rectangles get thinner and thinner).
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16. Anti Derivatives Area Under a Curve Riemann Sums Net Area Properties of the De…nite Integral
Example
f (x) = x2; a = 0, b = arbitrary
a = 0 nb/n
f(x) = x2
b/n 2b/n
f(x)L
L3b/n
divide into n rectangles
each rectangle has equal base-length =
b
n
.
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17. Anti Derivatives Area Under a Curve Riemann Sums Net Area Properties of the De…nite Integral
Example - continue
Base x
b
n
2b
n
3b
n
b =
nb
n
Height f (x)
b
n
2
2b
n
2
3b
n
2
b2
The sum of the areas of the rectangles
b
n
b
n
2
+
b
n
2b
n
2
+
b
n
3b
n
2
+ +
b
n
nb
n
2
=
b
n
3
12
+ 22
+ 32
+ + (n 1)2
+ n2
=
b
n
3 n
∑
i 1
i2
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18. Anti Derivatives Area Under a Curve Riemann Sums Net Area Properties of the De…nite Integral
Example - continue
n
∑
i=1
i2 =
n (n + 1) (2n + 1)
6
.
b3
n3
n (n + 1) (2n + 1)
6
=
b3 2n3 + 3n2 + n
6n3
=
2b3 +
3
n
+
1
n2
6
Take the limit as n ! ∞.
lim
n!∞
2b3 +
3
n
+
1
n2
6
=
b3
3
So the sum of the areas of the rectangles:
Z b
0
x2
dx =
b3
3
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19. Anti Derivatives Area Under a Curve Riemann Sums Net Area Properties of the De…nite Integral
Examples
f (x) = x. The area under the curve:
Z b
0
xdx =
b2
2
f (x) = 1. The area under the curve:
Z b
0
1dx =
b1
1
= b
In general, f (x) = xn. The area under the curve:
Z b
0
xn
dx =
bn+1
n + 1
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20. Anti Derivatives Area Under a Curve Riemann Sums Net Area Properties of the De…nite Integral
General Procedures for De…nite Integrals
Divide the base into n intervals with equal length ∆x.
x∆
ix
a b
( )xfy =
( )ixf
∆x =
b a
n
; xi = a + i∆x
The Riemann sum:
n
∑
i=1
f (xi ) ∆x :
Z b
a
f (x) dx = lim
n!∞
n
∑
i=1
f (xi ) 4x
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21. Anti Derivatives Area Under a Curve Riemann Sums Net Area Properties of the De…nite Integral
General Procedures for De…nite Integrals - continue
Note that: Z b
a
f (x) dx = lim
n!∞
n
∑
i=1
f (xi ) 4x
can also be written as
Z b
a
f (x) dx = lim
n!∞
b a
n
n
∑
i=1
f a +
i (b a)
n
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22. Anti Derivatives Area Under a Curve Riemann Sums Net Area Properties of the De…nite Integral
Example - continue
To evaluate
Z 1
0
x2dx : 4x =
1 0
n
=
1
n
; xi = 0 + i4x =
i
n
.
So, the de…nite integral is
Z 1
0
x2
dx = lim
n!∞
n
∑
i=1
f
i
n
1
n
= lim
n!∞
1
n
n
∑
i=1
f
i
n
= lim
n!∞
1
n
n
∑
i=1
i2
n2
= lim
n!∞
1
n3
n
∑
i=1
i2
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23. Anti Derivatives Area Under a Curve Riemann Sums Net Area Properties of the De…nite Integral
Example - continue
= lim
n!∞
1
n3
n
∑
i=1
i2
= lim
n!∞
1
n3
n (n + 1) (2n + 1)
6
= lim
n!∞
2n2 + 3n + 1
6n2
= lim
n!∞
2 +
3
n
+
1
n2
6
=
2
6
=
1
3
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24. Anti Derivatives Area Under a Curve Riemann Sums Net Area Properties of the De…nite Integral
Example
f (x) = x3 6x is a bounded function on [0, 3] . To evaluate the
Riemann sum with n = 6,
4x =
3 0
6
= 0.5
x1 = 0 + 0.5 = 0.5, x2 = 1.0, x3 = 1.5, x4 = 2.0, x5 = 2.5, x6 = 3.0.
So, the Riemann sum is
n
∑
i=1
f (xi ) 4x
=
1
2
[f (0.5) + f (1.0) + f (1.5) + f (2.0) + f (2.5) + f (3.0)]
=
1
2
( 2.875 5 5.625 4 + 0.625 + 9)
= 3.9375
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25. Anti Derivatives Area Under a Curve Riemann Sums Net Area Properties of the De…nite Integral
Example - continue
To evaluate the de…nite integral
Z 3
0
x3 6x dx : 4x =
3 0
n
=
3
n
; xi = 0 + i4x =
3i
n
.
So, the de…nite integral is
Z 3
0
x3
6x dx = lim
n!∞
n
∑
i=1
f (xi ) 4x
= lim
n!∞
n
∑
i=1
f
3i
n
3
n
= lim
n!∞
3
n
n
∑
i=1
"
3i
n
3
6
3i
n
#
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26. Anti Derivatives Area Under a Curve Riemann Sums Net Area Properties of the De…nite Integral
Example - continue
= lim
n!∞
3
n
n
∑
i=1
27
n3
i3 18
n
i
= lim
n!∞
3
n
"
27
n3
n
∑
i=1
i3 18
n
n
∑
i=1
i
#
= lim
n!∞
"
81
n4
n
∑
i=1
i3 54
n2
n
∑
i=1
i
#
= lim
n!∞
"
81
n4
n (n + 1)
2
2
54
n2
n (n + 1)
2
#
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27. Anti Derivatives Area Under a Curve Riemann Sums Net Area Properties of the De…nite Integral
Example - continue
= lim
n!∞
81
4
n4 + 2n3 + n2
n4
54
2
n2 + n
n2
= lim
n!∞
81
4
1 +
2
n
+
1
n2
27 1 +
1
n
= lim
n!∞
"
81
4
1 +
1
n
2
27 1 +
1
n
#
=
81
4
27 =
27
4
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28. Anti Derivatives Area Under a Curve Riemann Sums Net Area Properties of the De…nite Integral
Net Area
Geometrically the value of the de…nite integral represents the area
bounded by y = f (x) , the x axis and the ordinates at x = a and
x = b only if f (x) 0.
If f (x) is sometimes positive and sometimes negatives, the de…nite
integral represents the algebraic sum of the area above and below
the x axis (the net area).
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29. Anti Derivatives Area Under a Curve Riemann Sums Net Area Properties of the De…nite Integral
Area Under The Curve and Net Area
b
x∆
kxa
( )xfy =
y
x
If f (x) 0, the Riemann
sum
n
∑
k=1
f (xk ) 4x is the
sum of the areas of rectangles.
ba
y
x
( )xfy =
If f (x) 0, the Integral
Z b
a
f (x) dx is the area under
the curve from a to b.
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30. Anti Derivatives Area Under a Curve Riemann Sums Net Area Properties of the De…nite Integral
Area Under The Curve and Net Area
x
y
)(xfy =
+ +
-
ba
Z b
a
f (x) dx is the net area
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31. Anti Derivatives Area Under a Curve Riemann Sums Net Area Properties of the De…nite Integral
Extend Integration to the Case f < 0 - Example
Z 2π
0
sin xdx
x
y
Z 2π
0
sin xdx = ( cos x)j2π
0
= ( cos 2π) ( cos 0) = 1 + 1 = 0
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32. Anti Derivatives Area Under a Curve Riemann Sums Net Area Properties of the De…nite Integral
Total Distance and Net Distance
Total distance travelled:
Z b
a
jv (t)j dt
Net distance travelled: Z b
a
v (t) dt
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33. Anti Derivatives Area Under a Curve Riemann Sums Net Area Properties of the De…nite Integral
Monotonicity, Continuity and Integral
Theorem 2
Every monotonic function f on [a, b] is integrable.
Theorem 3
Every continuous function f on [a, b] is integrable.
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34. Anti Derivatives Area Under a Curve Riemann Sums Net Area Properties of the De…nite Integral
Properties of the De…nite Integral
Let f and g be integrable functions on [a, b], and c is a constant. Then,
1.
Z b
a
cdx = c (b a)
2.
Z a
a
f (x) dx = 0
3.
Z b
a
f (x) dx =
Z a
b
f (x) dx
4. cf is integrable and
Z b
a
cf (x) dx = c
Z b
a
f (x) dx.
5. f g is integrable and
Z b
a
(f g) (x) dx =
Z b
a
f (x) dx
Z b
a
g (x) dx.
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35. Anti Derivatives Area Under a Curve Riemann Sums Net Area Properties of the De…nite Integral
Properties of the De…nite Integral - continue
6.
Z b
a
f (x) dx =
Z c
a
f (x) dx +
Z b
c
f (x) dx provided that f is integral
on [a, c] and [c, b] . (works without ordering a, b, c )
7. (Estimation) If f (x) g (x) for x 2 [a, b] , then
Z b
a
f (x) dx
Z b
a
g (x) dx. (a < b )
8. jf j is integrable and
Z b
a
f (x) dx
Z b
a
jf (x)j dx.
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36. Anti Derivatives Area Under a Curve Riemann Sums Net Area Properties of the De…nite Integral
Example - Illustration of Property (6).
ex 1, x 0
Z b
0
ex dx
Z b
0
1dx
Z b
0
ex
dx = (ex
)jb
0 = eb
1
Z b
0
1dx = b
eb
1 + b, b 0
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37. Anti Derivatives Area Under a Curve Riemann Sums Net Area Properties of the De…nite Integral
Example - continue
Repeat:
ex 1 + x, x 0
Z b
0
ex dx
Z b
0
(1 + x) dx
Z b
0
ex
dx = (ex
)jb
0 = eb
1
Z b
0
(1 + x) dx = x +
x2
2
b
0
= b +
b2
2
eb
1 + b +
b2
2
, b 0
Repeat: Gives a good approximation of ex .
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