March 13, 2002
(1) Find the area of the region contained between the graphs of
The region is sketched below. Notice that the curves intersect at (0,0) and
(2,4), and is
the upper function on [0,2].
The area is thus (4-8/3)-(0-0/3)
= 4/3 square units.
(2) Consider region under the curve
between x = 1 and x = 2.
(a) The region is revolved around the x-axis. Find the volume of the resulting solid.
The cross section at x is a circle of radius 1/x, so its area is
A(x) = ππ
Thus the volume of the solid is
= π/2 cubic units.
(b) The region is revolved around the y-axis. Find the volume of the resulting solid.
Using the method of shells, we get V====
=4π-2π = 2π cubic units.