Math 142 WIR, copyright Angie Allen, Spring 2013
Math 142 Week-in-Review #8 (Sections 6.1, 6.2, and 6.3)
1. Find each of the following indefinite integrals:
a)
√
4x2 − x3 + π x − 7ex + 2 dx
Z b)
Z
e3x
dx
(e3x − 5)7
c)
Z
7x(x + 4)5 dx
d)
Z
e)
Z
!
√
√
3
12 + 5 x − 3x3 + 7 x2
− 6ex dx
x4
√
2x
dx
6−x
1
2
Math 142 WIR, copyright Angie Allen, Spring 2013
2. 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.
b) left endpoints. Sketch the graph and the rectangles, and show your formula.
c) midpoints. Sketch the graph and the rectangles, and show your formula.
Math 142 WIR, copyright Angie Allen, Spring 2013
3
3. 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 if necessary.)
4. Given
Z
2x
√
dx, what integral would be obtained if the appropriate u-substitution was made?
4
x+8
4
5. Compute
∑ 3i2 .
i=1
4
Math 142 WIR, copyright Angie Allen, Spring 2013
6. Write the formulas for a left, right, and general Riemann sum using sigma notation.
7. Find the function R(t) if
2
dR
= 3tet and R(0) = 4.
dt
5
Math 142 WIR, copyright Angie Allen, Spring 2013
8. The marginal cost function for a particular item is given by f (x) = −0.06x + 20 dollars per item. If the total
cost for producing 50 items is $1, 500, find the fixed cost associated with producing this item.
2
9. Find the most general antiderivative of the function f (x) = −x
1
9− 3
x
+
6
− 4π .
x
10. 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.
Math 142 WIR, copyright Angie Allen, Spring 2013
11. Find each of the following indefinite integrals:
2
a)
Z
e1/x
dx
x3
b)
Z
1
√ dx
x ln x
c)
Z
e2x − e−2x
dx
e2x + e−2x
d)
Z
16x ln(4x2 + 3)
dx
4x2 + 3
6
7
Math 142 WIR, copyright Angie Allen, Spring 2013
12. 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.
13. 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 formulas).
v
v=f(t)
4
3
2
1
1.5
t
4.5
3
14. 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 formula).
Time (s)
Velocity (ft/s)
0
25
5
31
10
35
15
43
8
Math 142 WIR, copyright Angie Allen, Spring 2013
15. Find
Z
x3 (x2 + 3)2 dx.
16. When finding
is made.
Z
2
(x + 4)e3x
+24x
dx, write the integral that would be obtained after the appropriate u-substitution