Math 173
Test 2
Name_______________________(5 pts)
1. It takes 7.5 lb of force to compress a 5-inch spring down to 4 inches. Assuming the spring actuation is linear, calculate
the work involved in compressing the spring to 3 inches long. (Show all work; integrate by hand; and express your
answer in conventional units.) (10 pts)
2. A pipe 5 ft long is more dense on one side than the other. The density is a function of its length: δ(x) = 10x2, where
density is measured in pounds per foot, and x is measured in feet. (Show all work; integrate by hand; and express your
answers in conventional units.)
a) Find the weight of the pipe. (5 pts)
b) Find the pipe’s center of gravity. (10 ps)
3. For the following consider the relation x2/100 + y2/100 = 1.
a) Properly parameterize the coordinates x and y, to
accommodate a circle of radius = 10. (5 pts)
b) Sketch a graph of the relation. (5 pts)
Math 173
c) Set up (but do NOT solve) the integral for arc length
using rectangular coordinates. (5 pts)
Test 2
Name_______________________(5 pts)
d) Set up (but do NOT solve) the integral for arc length
using the parametric coordinates. (5 pts)
c) Find the arc length of the circle by hand, using either
rectangular or parametric representation. YOU MUST
SHOW YOUR WORK. (10 pts)
4. For the following, consider the region in the x-y plane, bounded by f(x) = cosx, the x-axis, x = -π/2 and x = π/2.
a) Set up (but do not solve) the integral of volume by disk
rotation about the x-axis. (5 pts)
b) Set up (but do not solve) the volume by cylindrical shell
rotation about the x-axis. (5 pts)
c) Use your calculator to find the the volume of rotation
by either the disk or the cylindrical shell method. (5 pts)
d) Set up the surface area integral, and use your calculator
to find the surface area generated by revolving f(x) = cosx
from x = -π/2 to x = π/2 about the x-axis. (10 pts)
Math 173
Test 2
Name_______________________(5 pts)
5. The water tank shown below must be drained for maintenance. How long will it take a ½ hp motor, producing 275
ft-lbs per second, to pump the tank from the top, given the tank is half full of water? (You may use your calculator to
integrate, but you must show the important steps in solving this problem!) Note: Water weighs 62.3 pounds per
cubic foot. (15 pts)
10 ft
20 ft