Math 1312
D.Wilson
Name
Exam #1 Information
Our first exam will be given on Thursday, February 14.
The best way to study for the exam is to do the following:
• Go over the notes, graded homework, and all assigned problems from Sections 7.5 and
7.7. You may want to redo some of the problems.
• Redo the quizzes (solutions are on WebCT).
• Do the problems in this packet.
• Do these problems from the Chapter 7 Review: #2 - 6, 23 - 30, 37 - 44, 52 - 54
1. Consider the region bounded by the curves
1
y = ex , y = 1 − x, and x = 2.
2
(a) Find the exact area of the region.
(b) Find the exact volume of the solid generated when the region is rotated about
the vertical line x = 3.
2. A man stands on a platform that is 30 feet above the ground. He uses a rope of mass
density 0.5 lb/f t to pull up a bucket of water at a constant velocity. When the bucket
of water is on the ground, it weighs 50 pounds. But there is a hole at the bottom of
the bucket, so that water leaks out at a steady rate. When the bucket reaches the
platform, it weighs 25 pounds.
(a) Draw a picture related to this problem, and clearly label y = 0. Then find the
weight of the bucket in terms of its height.
(b) Find the total amount of work required to raise the bucket.
Estimate the answer, and include the units.
3. Consider the curve y = sin−1 (x) from x = 0 to x = 1.
Suppose that this curve is rotated about the y-axis to form an open tank. Assume that
the measurements along the x- and y-axes are in meters. If the tank is full of water,
find the work required to pump all the water over the top of the tank.
The density of water is 9, 800 N/m3 . Estimate the answer, and include the units.
4. A right circular cone has radius R and height H. Use horizontal slicing to find the
volume of the cone.
5. The density of oil in a circular oil slick on the surface of the ocean at a distance r
meters from the center of the slick is given by
δ(r) =
40
kg/m2 .
1 + r2
If the slick extends from r = 0 to r = 5,000 m, find the mass of oil in the slick.
Estimate the answer, and include the units.
6. Consider the region bounded by y = x2 − 3 and y = 0.
Suppose the region above is the vertical side of a tank that is 12 m long and full of
diesel fuel oil. Assume that the measurements along the x- and y-axes are in meters.
Find the force of the fuel on the vertical side of the tank.
The density of diesel fuel oil is 8, 330 N/m3 .
Estimate the answer, and include the units.
7. (a) Graph the region bounded by these equations. Label the intersection points.
√
y = x, y = 6 − x, and y = 0
(b) Set up (but do not evaluate) the integral that gives the exact area of the region.
(c) Set up (but do not evaluate) the integral that gives the volume of the solid generated when the region is rotated about the line x = −2.
(d) Set up (but do not evaluate) the integral that gives the volume of the solid generated when the region is rotated about the line y = 8.
8. (a) Graph the region bounded by these equations. Label the intersection points.
y = x2 − 2, and y = 1 − 2x
(b) Write an expression for ∆V of the 3D figure obtained when a vertical slice of the
region is rotated about the line y = −3.
(c) Write and evaluate the integral that gives the exact volume of the solid generated
when the region is rotated about the line y = −3.