Math 3B
Integral Examples
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Evaluate the definite integral below
4
(𝑥 3 +2)𝑑𝑥
0
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Evaluate the definite integral below
4
(𝑥 3 +2)𝑑𝑥
0
4
0
𝑥4
3
(𝑥 +2)𝑑𝑥 =
+ 2𝑥
4
4
0
(4)4
=
+ 2(4) −
4
0 4
+2 0
4
= 72
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Evaluate the definite integral below
5
(𝑒 3𝑥 )𝑑𝑥
1
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Evaluate the definite integral below
5
(𝑒 3𝑥 )𝑑𝑥
1
5
1
𝑒 3𝑥
3𝑥
(𝑒 )𝑑𝑥 =
3
5
1
𝑒 3(5)
𝑒3 1
=
−
3
3
=
1 15
(𝑒 − 𝑒 3 ) ≅ 1,089,666
3
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Evaluate the definite integral below
𝜋
cos(2𝑥)𝑑𝑥
−𝜋
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Evaluate the definite integral below
𝜋
cos(2𝑥)𝑑𝑥
−𝜋
𝜋
−𝜋
−sin(2𝑥)
cos(2𝑥)𝑑𝑥 =
2
𝜋
=
−𝜋
−sin(2𝜋)
− sin −2𝜋
−
2
2
=0
This integral is zero because cosine is an even function,
and the integral is over an even interval. The areas
above and below the axis cancel out.
Graph y=cos(2x) and see for yourself.
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Evaluate the definite integral below
1
1 − 𝑥 2 𝑑𝑥
0
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Evaluate the definite integral below
1
1 − 𝑥 2 𝑑𝑥
0
1
1 − 𝑥 2 𝑑𝑥 =
0
𝜋
4
𝑦=
1 − 𝑥2 → 𝑦2 = 1 − 𝑥2 → 𝑥2 + 𝑦2 = 1
Equation of a circle
At this point we don’t have the tools to be able to figure
out the antiderivative for this one. But we can use
geometry if we recognize the integrand as a circle of
radius 1. The integration region is just the first quadrant,
so the area is ¼ of the area of the circle.
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Below is a graph of a function. Call it g(t).
Draw a graph of the function defined by this definite integral:
𝑥
𝑓 𝑥 =
5
𝑔 𝑡 𝑑𝑡
4
0
g(t)
3
2
1
2
2
4
6
1
2
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Below is a graph of a function. Call it g(t).
Draw a graph of the function defined by this definite integral:
𝑥
𝑓 𝑥 =
5
𝑔 𝑡 𝑑𝑡
4
0
g(t)
3
g(t) is decreasing where
f(x) is concave down
2
g(t) is increasing where
f(x) is concave up
1
2
2
1
4
6
Peak of g(t) is
steepest point of f(x)
f(x)
2
Zero of g(t) is low
point of f(x)
Zero of g(t) is
high point of f(x)
10
5
2
2
4
6
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Find the derivative of the definite integral below
x
𝑡 3 sin(𝑒 𝑡 )𝑑𝑡
𝑔 𝑥 =
0
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Find the derivative of the definite integral below
x
𝑡 3 sin(𝑒 𝑡 )𝑑𝑡
𝑔 𝑥 =
0
𝑔′ 𝑥 = 𝑥 3 sin(𝑒 𝑥 )
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Find the derivative of the definite integral below
cos(𝑥 2 )
𝑡 3 sin(𝑒 𝑡 )𝑑𝑡
𝑔 𝑥 =
0
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Find the derivative of the definite integral below
cos(𝑥 2 )
𝑡 3 sin(𝑒 𝑡 )𝑑𝑡
𝑔 𝑥 =
0
𝑔′ 𝑥 = cos(𝑥 2 ) 3 sin(cos(𝑥 2 )) ∙ −2𝑥 ∙ 𝑠𝑖𝑛(𝑥 2 )
Chain rule
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Evaluate the definite integral below
ln(4)
3
(3𝑥 2 𝑒 𝑥 )𝑑𝑥
0
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