MATH 151 Engineering Math I, Spring 2014
JD Kim
Week15 Section 6.4
Section 6.4 The Fundamental Theorem of Calculus
Ex1) Compute the following definite integrals using the graph given.
1-1)
RA
f (x) dx
1-2)
RC
f (x) dx
1-3)
RB
f (x) dx
1-4)
RD
f (x) dx
1-5)
RA
f (x) dx
1-6)
RC
f (x) dx
0
0
A
0
C
C
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The Fundamental Theorem of Calculus
1. Part I If f is continuous on [a, b], then the function g defined by
Z x
g(x) =
f (t) dt
a
is continuous on [a, b] and differentiable on (a, b) and
g ′ (x) = f (x).
2. Part II If f is continous on [a, b], then
Z
b
a
f (x) dx = F (b) − F (a),
where F is an antiderivative of f .
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Rx
Ex1) If g(x) = 0 f (t)dt, where the graph of f (t) is given below, where 0 ≤ x ≤
10, evaluate g(0), g(3), g(6) and g(10). Where is g(x) increasing? What is the
maximum value of g(x)?
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Ex2) Find the derivative
Rx √
2-1) g(x) = −1 t3 + 1 dt
t2
dt
t2 + 1
2-2) g(x) =
R √x
2-3) g(x) =
R sin x cos t
dt
x2
t
1
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Ex3) Evaluate
R2 1
3-1) 1 2 dx
x
3-2)
R0
3-3)
R ln 6
3-4)
R 2 x6 − x2
dx
1
x7
3-5)
R2
0
(w 3 − 1)2 dw
3-6)
R0
|x2 − 1| dx
−1
(5x2 − 4x + 3) dx
ln 3
−2
8et dt
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Ex4) Evaluate
R π/2
0
f (x) dx where f (x) =



 cos x if 0 ≤ x < π/4


 sin x if π/4 ≤ x ≤ π/2
Ex5) Suppose an object is moving according to velocity v(t) = 3t − 5, 0 ≤ t ≤ 3.
Find the displacement and distance traveled during the first 3 seconds.
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