3473 MAC - FALL 2007 REVIEW PROBLEMS FOR TEST 1
1. Find the following indenite integrals.
a)
d)
g)
x2 cos(3x) dx
3x2 + 2x + 9
dx
(x + 1)(x2 + 4)
dx
x + x ln x
x tan1 x dx
b)
dx
x 1 4x2
1 x
dx
1+ x
e)
h)
c)
sin5 (4x) dx
f)
sec4 (5x) tan(5x) dx
4x2 9 dx
x
i)
2. The region R is bounded by the curves y = 2x, y = 1 + 3x2 and
the y -axis. Find the volume of the solid obtained by rotating R: (a)
about the x-axis; (b) about the y -axis.
3. The base of a solid is the region inside the curve x4 + y 2 = 1. Each
cross-section of the solid perpendicular to the x-axis is a square. What
is the volume of the solid?
4. A person using a spring-like exercise machine is able to stretch the
spring a distance of 2 feet from rest by expending units W1 of work.
If the spring had been stretched a distance of 3 feet instead, then the
work required would have been W2 units. Find the ratio W2 /W1 .
5. An aquarium is 4 m long, 1.5 m wide, and 1 m deep is full of water.
Find the work needed to pump half of the water out of the aquarium.
Leave your answer as a denite integral. DO NOT EVALUATE THE
INTEGRAL. You may assume that the density of water is 1000 kg/m3
and that the gravitational constant is g = 9.8.
6.
Derive the formula
7.
Derive the reduction formula
1
n1
cosn x dx = cosn1 x sin x +
n
n
1
V = a2 h
3
for the volume of a square pyramid using the cross-sectional method,
where h is the perpendicular height and a is length of one side of the
square base.
1
cosn2 x dx.