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Math 131 WIR, copyright Angie Allen
Math 131 Week-in-Review #9 (Sections 4.8, 5.1, and 5.2)
√
4 3
x + 3x7 8
√
1. Find the most general antiderivative of the function f (x) =
− .
5 4
x
x
2
2. Find the most general antiderivative of the function f (x) = −x
3. Find g if g ′′ (t) = 5et + 3 sint, g(0) = 0, and g(π ) = 0.
1
9− 3
x
+
6 sec x tan x
.
π
2
Math 131 WIR, copyright Angie Allen
4. Estimate the area under the graph of f (x) =
and
√
x + 4 from x = −3 to x = 1 using two subintervals (rectangles)
a) right endpoints. Sketch the graph and the rectangles, and show your formula (expanded).
b) left endpoints. Sketch the graph and the rectangles, and show your formula (expanded).
c) midpoints. Sketch the graph and the rectangles, and show your formula (expanded).
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Math 131 WIR, copyright Angie Allen
5. Filters at a water treatment plant become less effective over time, and hence the amount of pollution passing
through the filters increases over time. The rate at which pollution passes through the filters over a 35 day
period is given in the table below.
Day
Rate (kg/day)
0
3
7
5
14
8
21
15
28
25
35
45
Find upper and lower estimates of the amount of pollution passing through the filters during the 35 day
period.
6. Bob’s car is traveling at 40 mi/h when he hits the brakes, producing a constant deceleration of 23 ft/s2 . What
is the distance Bob’s car travels before coming to a stop?
Math 131 WIR, copyright Angie Allen
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7. Initially, a school of fish consists of 15 fish. The rate at which the school of fish grows is given by g(t) = 3et
fish per month, where t is measured in months and 0 ≤ t ≤ 12. How many fish will the school consist of
after 6 months? (Round to the nearest integer.)
8. Find an expression for the area of the region that lies under the graph of f (x) = ex + 2 on −2 ≤ x ≤ 1 as a
limit. Do not evaluate the limit. Then, estimate the area by taking the sample points to be midpoints and
using 2 subintervals.
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Math 131 WIR, copyright Angie Allen
9. A stone was dropped off a cliff and hit the ground with a speed of 110 ft/s. What is the height of the cliff?
10. 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 131 WIR, copyright Angie Allen
11. Express
Z π
(sin 3x + 4 ln x) dx as a limit of a right sum.
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12. Write the following as a single integral:
Z 5
f (x) dx +
5
−5
13. Find bounds to estimate the value of
Z 8
Z 8
0
f (x) dx −
(x2 − 10x + 28) dx.
Z −3
−5
f (x) dx
Math 131 WIR, copyright Angie Allen
14. Use the graph of f (x) below to answer parts a)-f).
a) Find
Z 4
f (x) dx.
Z 0
f (x) dx.
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 131 WIR, copyright Angie Allen
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15. Write the formulas for a left, right, and general Riemann sum using sigma notation.
16. When approximating the area under f (x) = −0.3x2 + 10x on the interval [6, 8] using a midpoint sum with 4
rectangles, what is the area of the third rectangle? Remember to also show your formula.
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Math 131 WIR, copyright Angie Allen
17. Write a mathematical expression (i.e., formula) representing the area of the second rectangle when finding
a left Riemann sum of a function f on the interval [a, b] using n rectangles.
18. An object travels with a velocity function shown below (in feet/second), where t is time in seconds. Find
an upper and lower estimate of the distance traveled by the object (remember to also show your expanded
formulas).
v
v=f(t)
4
3
2
1
1.5
t
4.5
3
19. The table below shows the velocity (ft/s) of an object every five seconds over a 15 second time interval.
Estimate the total distance (in feet) the object travels over the 15 second time interval by finding an upper
estimate (remember to also show your expanded formula).
Time (s)
Velocity (ft/s)
0
25
5
31
10
35
15
43