Math 1210-1
HW 14
Due Wednesday April 21, 2004
Please show all of your work and box your answer. Be sure to write in complete sentences when appropriate.
Volumes of Solids
1. Use your favoite method for determining the volume of the following solids.
(a)
(b)
(c)
(d)
Region
Region
Region
Region
bounded
bounded
bounded
bounded
by
by
by
by
x = y 2 , y = 2, x = 0, y = 0 rotated about the line y = 2.
√
x = 2y + 1, y = 2, x = 0 rotated about the line y = 3.
y = x3 , x = 1, y = −1 rotated about the line y = −1.
√
y = x, x = 1, y = 0 rotated about the line x = 1
2. Most states expect to run out of space for their garbage soon. In New York, solid garbage is packed
into pyramid-shaped dumps with square bases. (The largest such dump is on Staten Island.) A small
community has a dump with a base length of 100 yards. One yard vertically above the base, the length
of the side parallel to the base is 99 yards; the dump can be built up to a vertical height of 20 yards.
(The top of the pyramid is never reached.) If 65 cubic yards of garbage arrive at the dump every day,
how long will it be before the dump is full?
Length of a Plane Curve; Surface Area
3. Sketch each of the following curves and find their lengths.
(a) y = 9x3/2 between x = 1 and x = 4
x4 + 1
(b) y =
between x = 1 and x = 2
2x
(c) 2xy 3 − y 4 = 1 between y = 1 and y = 2
(d) x = t2 + 1, y = t3 + 2t, 0 ≤ t ≤ 2
(e) x = 2 sin t − 1, y = 2 cos t + 3, 0 ≤ t ≤
π
4.
4. The curve in Problem 3(e) is revolved around the x-axis. What is the area of the resulting surface?
5. A point P on the rim of a wheel of radius a is initially at the origin. As the wheel rolls to the
right along the x-axis, P traces out a curve called a cycloid (I encourage you to set up a simple
experiment at home to see what this curve looks like). The cycloid is given by the parametric equations
x = a(θ − sin θ), y = a(1 − cos θ) where the parameter θ is the angle through which the wheel has
rolled.
Find the length of one arch of the cycloid, by first showing that
2 2
dx
θ
dy
2
2
.
+
= 4a sin
dθ
dθ
2
6. Suppose that the wheel of Problem 5 turns at a constant rate ω =
(a) Show that the speed
ds
of P along the cycloid is
dt
ds
ωt = 2aω sin .
dt
2
dθ
, where t is time. Then θ = ωt.
dt
(b) When is the speed a maximum and when is it a minimum?
(c) Explain why a bug on a wheel of a car going 60 miles per hour is itself sometimes traveling at 120
miles per hour.
7. Find the area of the surface generated by revolving the given curve about the x-axis.
√
(a) y = 16 − x2 , −1 ≤ x ≤ 2
(b) y = 13 x3 ,
1≤x≤2
(c) x = 2t, y = t3 ,
2
0≤t≤1
(d) x = 1 − t , y = t,
0≤t≤1