Example 6 __
Find the volume of a sphere of radius r.
SOLUTION: If the region bounded by a semicircle (with center O and radius r)
and its diameter is revolved about the x-axis, the solid of revolution obtained is a
sphere of radius r, as seen in Figure N7–12.
Figure N7–12The volume ΔV of a typical disk is given by ∆V = πy^2 Δx. The equation of the
circle is x^2 + y^2 = r^2 . To find the volume of the sphere, we form a Riemann Sum
whose limit as n becomes infinite is a definite integral. Then,
.
Example 7 __
Find the volume of the solid generated when the region bounded by y = x^2 , x = 2,
and y = 0 is rotated about the line x = 2 as shown in Figure N7–13.