Math 166
Study Guide 1
Exam 1 on September 17, 2015
This is a short study guide for Exam 1. The exam will cover Chapter 6. It will consist of 6 problems,
one from each section. The first section of this guide lists some of the formulas we’ve seen in Chapter 6.
The second section contains some practice problems. As you go through them, I recommend you first make
sure you know how to set up all of the integrals that arise in these problems before trying to compute them.
The problems are not meant to be a comprehensive list of every nuanced detail, but rather a reminder of
the kinds of problems we’ve seen.
1
Formulas
1.1
Disk method
Suppose a solid of revolution S consists of disks of radius R(x) generated by revolving vertical strips
around a horizontal axis of revolution for a ≤ x ≤ b. Then the volume of S is given by
b
Z
2
π [R(x)] dx
a
If S consists of disks of radius R(y) generated by revolving horizontal strips around a vertical axis
of revolution for a ≤ y ≤ b. Then the volume V of S is given by
Z
b
2
π [R(y)] dy
a
1.2
Washer method
Suppose a solid of revolution S consists of washers with outer radius R(x) and inner radius r(x) generated
by revolving vertical strips around a horizontal axis of revolution for a ≤ x ≤ b. Then the volume V of S
is given by
Z b 2
2
π [R(x)] − [r(x)] dx
a
If S consists of washers with outer radius R(y) and inner radius r(y) generated by revolving horizontal
1
strips around a vertical axis of revolution for a ≤ y ≤ b. Then the volume V of S is given by
b
Z
2
2
π [R(y)] − [r(y)] dy
a
1.3
Shell method
Suppose a solid of revolution S consists of cylindrical shells with radius r(x) and height h(x) generated by
revolving vertical strips around a vertical axis of revolution for a ≤ x ≤ b. Then the volume of S is give
by
Z b
2πr(x)h(x) dx
a
Suppose a solid of revolution S consists of cylindrical shells with radius r(y) and height h(y) generated
by revolving horizontal strips around a horizontal axis of revolution for a ≤ y ≤ b. Then the volume of
S is give by
Z b
2πr(y)h(y) dy
a
1.4
Arc length
If f 0 is continuous on [a, b] then the arc length of the curve y = f (x) for a ≤ x ≤ b is given by
Z
b
p
1 + f 0 (x)2 dx
a
1.5
Surface area
If f (x) ≥ 0 for all x in [a, b] and f 0 is continuous on [a, b] then the area of the surface generated by revolving
y = f (x), a ≤ x ≤ b about the x-axis is given by
Z
b
p
2πf (x) 1 + f 0 (x)2 dx
a
1.6
Work done pumping fluid
Suppose that a tank contains a fluid of weight-density w from y = a to y = b on the y-axis. Suppose that a
thin horizontal slice of fluid at level y has volume A(y) dy. Then the work done in pumping the fluid to a
height L is
Z b
w · A(y) · (L − y) dy
a
1.7
Fluid force against vertical flat plate
Suppose that a plate submerged vertically in fluid of weight-density w runs from y = a to y = b on the
y-axis. Let L(y) be the length of the horizontal strip measured from left to right along the surface of the
plate at level y. Then the force exerted by the fluid against one side of the plate is
Z
b
w · (strip depth) · L(y) dy
a
1.8
Center of mass
The center of mass of a thin plate in the xy-plane is the point (x, y) given by
x=
R
e dm
Rx
,
dm
y=
R
e dm
Ry
dm
where (e
x, ye) is the center of mass of a typical strip and dm is the mass of a typical strip. Note that if the
plate is divided into vertical strips, then x̃, ỹ, and dm are functions of x. If the plate is divided into
horizontal strips, then x̃, ỹ, and dm are functions of y.
2
Problems
Here are some practice problems. As you go through them, I recommend you first make sure you know how
to set up all of the integrals that arise in these problems before trying to compute them. This list is not
meant to be a comprehensive list of every nuanced detail, but rather a reminder of the kinds of problems
we’ve seen. The exam will be six problems, all similar to homework.
1. Consider a solid S which lies between planes perpendicular to the x-axis at x = −1 and x = 1. The
cross-sections perpendicular to the x-axis are circular disks whose diameters run from the parabola
y = x2 to the parabola y = 2 − x2 . Find the volume of S.
√
2. Consider a solid S whose base is the region bounded by the graphs of y = x and y = x/2. The
cross-sections perpendicular to the x-axis are isosceles triangles of height 6. Find the volume of S.
√
3. Let R be the region bounded by y = 9 − x2 , y = 0, x = −2, and x = 3. Find the volume of the solid
formed by revolving R about the x-axis.
4. Let R be the region bounded by y = ex−1 , y = 0, x = 1, and x = 3. Find the volume of the solid
formed by revolving R about the x-axis.
√
5. Let R be the region bounded by x = 2 y, y = 4, and x = 0. Find the volume of the solid formed by
revolving R about the y-axis.
√
6. Let R be the region bounded by x = y, x = −y, and y = 2. Find the volume of the solid formed by
revolving R about the x-axis.
7. Let R be the region bounded by the line y = 4x and the parabola y = 4x2 . Find the volume of the
solid formed by revolving R about the y-axis.
8. Let R be the region bounded by x = y 2 , y = 1, and x = 0. Find the volume of the solid formed by
revolving R about the x-axis.
9. Find the length of the curve y = (x4 + 3)/(6x) between x = 1 and x = 3.
10. Find the length of the curve x = y 3 /6 + 1/(2y) from y = 2 to y = 3.
√
11. Let C be the curve y = 25 − x2 for −2 ≤ x ≤ 3. Find the area of the surface formed by revolving C
about the x-axis.
12. Let C be the curve x = y 3 /3 for 0 ≤ y ≤ 1. Find the area of the surface formed by revolving C about
the y-axis.
13. A force of 0.6 newtons is required to keep a spring with a natural length of 0.08 meters compressed
to a length of 0.07 meters. Find the work done in compressing the spring from its natural length to a
length of 0.06 meters.
14. Consider a trough filled with water (weight density 62.4 pounds per cubic foot) that is 10 feet long
with a vertical cross section in the shape of a regular trapezoid. The dimensions on on the trapezoid
are as follows: the upper and lower bases are of lengths 6 and 3 feet respectively, and the height is 4
feet. Find the work done in pumping the water from this tank to a height 5 feet above the top of the
tank.
15. A conical tank with height 6 feet and radius 2 feet is filled with a liquid with weight density 57 pounds
per cubic foot. The tank is oriented to that the vertex of the cone is at the bottom and the circle with
radius 2 is at the top. Find the work done in pumping the liquid from the tank to a height 2 feet above
the tank.
16. Let R be an equilateral triangle with side length 6 feet. Suppose R is submerged in a tank of water so
that one of its flat sides is at the water line. Find the total force exerted on R by the water.
17. A semicircular plate 2 feet in diameter is place into freshwater with its diameter facing upward, 2 feet
below the surface. Find the force exerted by the water on one side of the plate.
18. Let R be the region bounded by y = 31 x2 , y = 0, and x = 4. Find the centroid of R.
19. Let R be the regions bounded by y = x2 and y = x + 6. Find the centroid of R.