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Math 142 WIR, copyright Angie Allen, Spring 2013
Math 142 Week-in-Review #9 (Sections 6.4, 6.5, and 6.6)
1. Write the following as a single integral:
Z 5
f (x) dx +
5
−5
2. Find bounds to estimate the value of
Z 8
0
3. Evaluate
√
Z A 2
2
t + 4t − t 3
3
t3
Z 8
f (x) dx −
Z −3
−5
(x2 − 10x + 28) dx.
dt, where A is a constant and A > 3.
f (x) dx
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Math 142 WIR, copyright Angie Allen, Spring 2013
4. Use the graph of f (x) below to answer parts a)-f).
a) Find
Z 4
f (x) dx.
Z 0
f (x) dx.
(Courtesy of Joe Kahlig)
0
b) Find
4
c) Find
Z 10
f (x) dx.
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d) Find
Z 4
f (x) dx.
Z 9
f (x) dx.
4
e) Find
2
f) Find the area between the function and the x axis from x = 0 to x = 10.
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Math 142 WIR, copyright Angie Allen, Spring 2013
5. Find the area bounded by the curves f (x) = x2 and g(x) = −x + 4.
6. Estimate
Z 6
−2
(3x − x2 ) dx using a Riemann sum with n = 4 and the Midpoint Rule. What does your answer
represent? What does the definite integral represent?
7. If
Z B
A
f (x) dx = −15 and
Z B
A
[4 f (x) − 3g(x)] dx = 62, find
Z A
B
g(x) dx.
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Math 142 WIR, copyright Angie Allen, Spring 2013
8. When a cup of hot chocolate is bought its temperature is 175◦ F. The outside temperature is 50◦ F, and the rate
at which the hot chocolate is cooling is given by c(x) = −10e−0.08x degrees per minute, where x is measured
in minutes and x ≥ 0.
a) What is the temperature of the hot chocolate after 1 hour?
b) Find the change in the temperature during the first 6 minutes after the hot chocolate is bought, and interpret
your answer.
c) Find the average temperature between 5 and 10 minutes after the hot chocolate is bought.
9. Let G(x) be an antiderivative of g(x). If
Z 3
0
g(x) dx = 15 and G(3) = 7, find G(0).
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Math 142 WIR, copyright Angie Allen, Spring 2013
10. Find
Z A
2
x4
dx.
x5 + 5
11. Find the area bounded by the curves y = 0.6x + 3 and y = x2 − 3x + 1 on the interval [−3, 2].
12. Shade the area on the graph below which is represented by
Z 2
g(x) dx +
2
−3
Ramsey)
y
g(x)
x
−3
f(x)
2
5
Z 4
[ f (x) − g(x)] dx.
(Courtesy of Heather
Math 142 WIR, copyright Angie Allen, Spring 2013
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13. Find the derivative of the following functions. Simplify your answers.
a) y =
Z t2
ln(q + 3) dq
3
b) y =
Z √x 4x
2
et + 5t dt
14. Bootsie’s Bandana Company’s profit is given by P(x) = −x2 + 200x − 300 dollars, where x is the number of
bandanas produced and sold each month.
a) Find the average profit when the number of bandanas sold each month is between 170 and 190.
b) Find the average marginal profit when the number of bandanas sold each month is between 170 and 190.
c) Find the change in profit when the number of bandanas sold each month increases from 170 to 190.