Math 1452 Final Exam Fall 2022
Calculators are not allowed on this exam. Work all questions completely. Show all work as described in class.
Copyright 2022 Dept of Mathematics and Statistics, Texas Tech University. Unauthorized reproduction prohibited.
1. Consider the region bounded by the y-axis and the functions y = x2 and y = 3x.
Set up (but do not evaluate) integrals to find
(a) The area of this region.
(b) The volume of the solid generated by rotating this region about the y-axis using
shells.
(c) The volume of the solid generated by rotating this region about the vertical line
x = ?5 using washers.
2. Set up (but do not evaluate) an integral to find the arc length of the graph of
y = ln(x), 1 ? x ? 5.
3. Consider a cylindrical tank with radius 7 f t and height 12 f t. If it is filled with a fluid
weighing 45 lb/f t3 , set up an integral to find the work required to pump all the fluid
out of the tank.
4. Evaluate the following integrals.
Z ?
4 ? x2 dx
(a)
Z
(c)
x cos(x) dx
Z ?
5. Does
3
Z
9x
dx
(x ? 1)(x + 2)
?
Z
sec( x)
?
(d)
dx
x
(b)
1
dx converge? Explain completely.
x(ln(x))4
6. Indicate if the following series converge or diverge. You must identify all the tests you
use and show all the work needed to apply them.
k
?
?
X
X
1
3k+1
(a)
2+ 2
(b)
k
2k
k=1
k=2
?
X
4k
(c)
4
k +1
k=1
(d)
?
X
5k
k=0
k!
7. Find the Maclaurin series for f (x) = 3×2 cos(2x). Hint: If you know the Maclaurin series
for cos(x), you can use it to find the series for f .
8. Let u =?2, ?1, 3? and v= ?0, 3, 1?.
(a) Find u ? 2v.
(b) Find ||v||.

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