benginning calculus lecture notes 12 - anti derivatives indefinite and definite integrals
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Beginning Calculus- Antiderivatives and The Definite Integrals -
Shahrizal Shamsuddin Norashiqin Mohd Idrus
Department of Mathematics,FSMT - UPSI
(LECTURE SLIDES SERIES)
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Anti Derivatives Area Under a Curve Riemann Sums Net Area Properties of the Definite Integral
Learning Outcomes
Use substitution and advanced guessing methods to evaluate antiderivatives.
Compute Riemann Sums.
Compute areas under the curve and net areas.
State and apply properties of the definite integrals.
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Anti Derivatives Area Under a Curve Riemann Sums Net Area Properties of the Definite Integral
Anti Derivatives
G (x) =∫g (x) dx
G (x) is called the anti derivative of g , or the indefinite integralof g .
G ′ (x) = g (x)
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Anti Derivatives Area Under a Curve Riemann Sums Net Area Properties of the Definite Integral
Example
∫sin xdx = − cos x + C
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Anti Derivatives Area Under a Curve Riemann Sums Net Area Properties of the Definite Integral
Example
∫xadx =
xa+1
a+ 1+ C , for a 6= −1
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Anti Derivatives Area Under a Curve Riemann Sums Net Area Properties of the Definite Integral
Example
∫ dxx= ln |x |+ C
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Anti Derivatives Area Under a Curve Riemann Sums Net Area Properties of the Definite Integral
More Examples
∫sec2 xdx = tan x + C∫ dx√1− x2
= sin−1 x + C∫ dx1+ x2
= tan−1 x + C∫exdx = ex + C
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Anti Derivatives Area Under a Curve Riemann Sums Net Area Properties of the Definite Integral
Uniqueness of anti derivatives up to a constant
Theorem 1
If F ′ = G ′, then F (x) = G (x) + C .
Proof.
Suppose F ′ = G ′. Then,
(F − G )′ = F ′ − G ′ = 0F (x)− G (x) = C
⇒ F (x) = G (x) + C
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Anti Derivatives Area Under a Curve Riemann Sums Net Area Properties of the Definite Integral
Method of Substitution - For Differential Notation
∫x3(x4 + 2
)5dx
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Anti Derivatives Area Under a Curve Riemann Sums Net Area Properties of the Definite Integral
Example
∫ xdx√1+ x2
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Anti Derivatives Area Under a Curve Riemann Sums Net Area Properties of the Definite Integral
Example
∫e6xdx
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Anti Derivatives Area Under a Curve Riemann Sums Net Area Properties of the Definite Integral
Example - Advanced Guessing
∫xe−x
2dx
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Anti Derivatives Area Under a Curve Riemann Sums Net Area Properties of the Definite Integral
Example
∫ dxx ln x
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Anti Derivatives Area Under a Curve Riemann Sums Net Area Properties of the Definite Integral
Area Under a Curve
b
( )dxxfb
a∫
a
( )xfy =
Area under a curve =∫ ba f (x) dx
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Anti Derivatives Area Under a Curve Riemann Sums Net Area Properties of the Definite Integral
Area Under a Curve
To compute the area under a curve:
b
L
a
1 Divide into n rectangles2 Add up the areas3 Take the limit as n→ ∞ (the rectangles get thinner and thinner).
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Anti Derivatives Area Under a Curve Riemann Sums Net Area Properties of the Definite 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 =bn.
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Anti Derivatives Area Under a Curve Riemann Sums Net Area Properties of the Definite Integral
Example - continue
Base xbn
2bn
3bn
· · · b =nbn
Height f (x)(bn
)2 (2bn
)2 (3bn
)2· · · b2
The sum of the areas of the rectangles(bn
)(bn
)2+
(bn
)(2bn
)2+
(bn
)(3bn
)2+ · · ·+
(bn
)(nbn
)2=
(bn
)3 (12 + 22 + 32 + · · ·+ (n− 1)2 + n2
)=
(bn
)3 n
∑i−1
i2
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Anti Derivatives Area Under a Curve Riemann Sums Net Area Properties of the Definite Integral
Example - continuen
∑i=1
i2 =n (n+ 1) (2n+ 1)
6.
b3
n3
(n (n+ 1) (2n+ 1)
6
)=
b3(2n3 + 3n2 + n
)6n3
=2b3 +
3n+1n2
6Take the limit as n→ ∞.
limn→∞
2b3 +3n+1n2
6=b3
3So the sum of the areas of the rectangles:∫ b
0x2dx =
b3
3
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Anti Derivatives Area Under a Curve Riemann Sums Net Area Properties of the Definite Integral
Examples
f (x) = x . The area under the curve:∫ b0xdx =
b2
2
f (x) = 1. The area under the curve:∫ b01dx =
b1
1= b
In general, f (x) = xn . The area under the curve:∫ b0xndx =
bn+1
n+ 1
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Anti Derivatives Area Under a Curve Riemann Sums Net Area Properties of the Definite Integral
General Procedures for Definite Integrals
Divide the base into n intervals with equal length ∆x .
x∆
ixa b
( )xfy =
( )ixf
∆x =b− an
; xi = a+ i∆x
The Riemann sum:n∑i=1
f (xi )∆x :
∫ baf (x) dx = lim
n→∞
n
∑i=1
f (xi )4x
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Anti Derivatives Area Under a Curve Riemann Sums Net Area Properties of the Definite Integral
General Procedures for Definite Integrals - continue
Note that: ∫ baf (x) dx = lim
n→∞
n
∑i=1
f (xi )4x
can also be written as∫ baf (x) dx = lim
n→∞
(b− an
) n
∑i=1
f(a+
i (b− a)n
)
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Anti Derivatives Area Under a Curve Riemann Sums Net Area Properties of the Definite Integral
Example - continue
To evaluate∫ 10x2dx : 4x = 1− 0
n=1n; xi = 0+ i4x =
in.
So, the definite integral is∫ 10x2dx = lim
n→∞
n
∑i=1
f(in
)(1n
)= lim
n→∞
(1n
) n
∑i=1
f(in
)= lim
n→∞
(1n
) n
∑i=1
i2
n2
= limn→∞
(1n3
) n
∑i=1
i2
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Anti Derivatives Area Under a Curve Riemann Sums Net Area Properties of the Definite Integral
Example - continue
= limn→∞
(1n3
) n
∑i=1
i2
= limn→∞
(1n3
)(n (n+ 1) (2n+ 1)
6
)= lim
n→∞
2n2 + 3n+ 16n2
= limn→∞
2+3n+1n2
6=26=13
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Anti Derivatives Area Under a Curve Riemann Sums Net Area Properties of the Definite Integral
Example
f (x) = x3 − 6x is a bounded function on [0, 3] . To evaluate theRiemann sum with n = 6,
4x = 3− 06
= 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
=12[f (0.5) + f (1.0) + f (1.5) + f (2.0) + f (2.5) + f (3.0)]
=12(−2.875− 5− 5.625− 4+ 0.625+ 9)
= −3.9375
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Anti Derivatives Area Under a Curve Riemann Sums Net Area Properties of the Definite Integral
Example - continue
To evaluate the definite integral∫ 30
(x3 − 6x
)dx : 4x = 3− 0
n=3n; xi = 0+ i4x =
3in.
So, the definite integral is∫ 30
(x3 − 6x
)dx = lim
n→∞
n
∑i=1
f (xi )4x
= limn→∞
n
∑i=1
f(3in
)(3n
)
= limn→∞
(3n
) n
∑i=1
[(3in
)3− 6
(3in
)]
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Anti Derivatives Area Under a Curve Riemann Sums Net Area Properties of the Definite Integral
Example - continue
= limn→∞
(3n
) n
∑i=1
[27n3i3 − 18
ni]
= limn→∞
(3n
)[(27n3
) n
∑i=1
i3 −(18n
) n
∑i=1
i
]
= limn→∞
[(81n4
) n
∑i=1
i3 −(54n2
) n
∑i=1
i
]
= limn→∞
[(81n4
)(n (n+ 1)
2
)2−(54n2
)(n (n+ 1)
2
)]
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Anti Derivatives Area Under a Curve Riemann Sums Net Area Properties of the Definite Integral
Example - continue
= limn→∞
[(814
)(n4 + 2n3 + n2
n4
)−(542
)(n2 + nn2
)]= lim
n→∞
[(814
)(1+
2n+1n2
)− 27
(1+
1n
)]= lim
n→∞
[(814
)(1+
1n
)2− 27
(1+
1n
)]
=814− 27 = −27
4
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Anti Derivatives Area Under a Curve Riemann Sums Net Area Properties of the Definite Integral
Net Area
Geometrically the value of the definite integral represents the areabounded by y = f (x) , the x−axis and the ordinates at x = a andx = b only if f (x) ≥ 0.If f (x) is sometimes positive and sometimes negatives, the definiteintegral represents the algebraic sum of the area above and belowthe x−axis (the net area).
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Anti Derivatives Area Under a Curve Riemann Sums Net Area Properties of the Definite Integral
Area Under The Curve and Net Area
b
x∆
kxa
( )xfy =
y
x
If f (x) ≥ 0, the Riemann
sumn
∑k=1
f (xk ) · 4x is the
sum of the areas of rectangles.
ba
y
x
( )xfy =
If f (x) ≥ 0, the Integral∫ baf (x) dx is the area under
the curve from a to b.
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Anti Derivatives Area Under a Curve Riemann Sums Net Area Properties of the Definite Integral
Area Under The Curve and Net Area
x
y
)(xfy =+ +
ba
∫ baf (x) dx is the net area
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Anti Derivatives Area Under a Curve Riemann Sums Net Area Properties of the Definite Integral
Extend Integration to the Case f < 0 - Example
∫ 2π
0sin xdx
x
y
∫ 2π
0sin xdx = (− cos x)|2π
0
= (− cos 2π)− (− cos 0) = −1+ 1 = 0
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Anti Derivatives Area Under a Curve Riemann Sums Net Area Properties of the Definite Integral
Total Distance and Net Distance
Total distance travelled: ∫ ba|v (t)| dt
Net distance travelled: ∫ bav (t) dt
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Anti Derivatives Area Under a Curve Riemann Sums Net Area Properties of the Definite 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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Anti Derivatives Area Under a Curve Riemann Sums Net Area Properties of the Definite Integral
Properties of the Definite Integral
Let f and g be integrable functions on [a, b], and c is a constant. Then,
1.∫ bacdx = c (b− a)
2.∫ aaf (x) dx = 0
3.∫ baf (x) dx = −
∫ abf (x) dx
4. cf is integrable and∫ bacf (x) dx = c
∫ baf (x) dx .
5. f ± g is integrable and∫ ba(f ± g) (x) dx =
∫ baf (x) dx ±
∫ bag (x) dx .
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Anti Derivatives Area Under a Curve Riemann Sums Net Area Properties of the Definite Integral
Properties of the Definite Integral - continue
6.∫ baf (x) dx =
∫ caf (x) dx +
∫ bcf (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 ∈ [a, b] , then∫ baf (x) dx ≤
∫ bag (x) dx . (a < b )
8. |f | is integrable and∣∣∣∣∫ ba f (x) dx
∣∣∣∣ ≤ ∫ ba |f (x)| dx .
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Anti Derivatives Area Under a Curve Riemann Sums Net Area Properties of the Definite Integral
Example - Illustration of Property (6).
ex ≥ 1, x ≥ 0∫ b0exdx ≥
∫ b01dx
∫ b0exdx = (ex )|b0 = eb − 1∫ b01dx = b
eb ≥ 1+ b, b ≥ 0
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Anti Derivatives Area Under a Curve Riemann Sums Net Area Properties of the Definite Integral
Example - continue
Repeat:ex ≥ 1+ x , x ≥ 0∫ b
0exdx ≥
∫ b0(1+ x) dx
∫ b0exdx = (ex )|b0 = eb − 1∫ b
0(1+ x) dx =
(x +
x2
2
)∣∣∣∣b0= b+
b2
2
eb ≥ 1+ b+ b2
2, b ≥ 0
Repeat: Gives a good approximation of ex .
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