Math 1230: Calculus 2
Homework #7 Due Monday, May 18
Read: Section 6.2 & 6.3
Problem 1 Take the area under the graph of y=1
xpfrom x= 1 to x=∞,
and revolve it around the x-axis. For which values of pis the volume of the
resulting solid finite?
Solution The volume of the slice at position xis
volume = π1
xp2
dx
so the total volume is
Z∞
1
π1
xp2
dx =πZ∞
1
1
x2pdx.
We know from class that this integral converges when 2p > 1 and that it
diverges otherwise. We conclude that the volume is finite when p > 1
2.2
Problem 2 Take the same region as in Problem 1 and revolve it around the
y-axis. For which values of pdoes that solid have finite volume?
Solution There are two ways to go about this one.
•Method 1: Washers. The slice at position yhas volume
volume = πy−1
p2
−12dy
so the total volume is
Z1
0
πy−1
p2
−12dy =πZ1
0
y−2
pdy −π.
We know from Quiz 4, Problem 2 (in the future!) that this integral
converges when −2
p>−1, and diverges otherwise. Thus the volume is
finite for p > 2.
