# Page 1 Sections 6.4: The Fundamental Theorem of Calculus

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Sections 6.4: The Fundamental Theorem of Calculus
Definition:
Z The indefinite integral of f is used to indicate the process of finding the antiderivative
of f . i.e.
f (x) dx = F (x) + C
Example: Compute.
A)
Z
2x5 + 7x + 4 dx =
B)
Z
3x2 da =
The Fundamental Theorem of Calculus, Part 2 If f is continuous on [a, b], then
Zb
a
f (x) dx = F (b) − F (a),
where F is any antiderivative of f .
Example: Compute the following.
A)
Z5
3x2 + 4x + 2 dx =
1
B)
Z4
0
3x + 8e4x dx =
C)
Z5
1
dx =
x2
−2
D)
Z3
0
E)
|x2 − 4| dx =
Z9 √
4
1
x− √
x
2
dx =
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The Fundamental Theorem of Calculus, Part 1 If f is continuous on [a, b], then the function g
defined by
g(x) =
Zx
f (t) dt,
a
with a ≤ x ≤ b
is continuous on [a, b] and is differentiable on (a, b), and g′ (x) = f (x)
Example: Find g′ (x).
A) g(x) =
Zx
t2 + 1 dt
Zx2
tan3 (t) dt
a
B) g(x) =
4
C) g(x) =
Z2
ln(u) du
x3
D) g(x) =
xZ3 +1
x2
u5 + 2 du
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Example: Use the graph of f ′ (x) to answer these questions.
6
4
2
20
5
−2
10
25
30
15
−4
A) Which is larger? f (15) or f (25)
B) If f (5) = 30, find f (20).
Example: Define g(a) by g(a) =
Za
f (x) dx where f (x) is the graph given below. Find the intervals
0
where g(a) is increasing. If possible, give the values of the absolute maximum and absolute minimum.
6
4
2
15
−2
−4
5
10
20 25 30 35 40