Question

A sphere of radius R is centered at the origin. A plane intersects this sphere at a distance a from its center such that a<R. determine the volume of the smaller of the regions bound by the sphere and the plane . ( you may use Cartesian coordinates or spherical polar coordinates.)

Answer 1

QUESTION

A sphere of radius $R$ is centered at the origin. A plane intersects this sphere at a distance $a$ from its center such that $a < R$ . Determine the volume of the smaller of the regions bound by the sphere and the plane. (you may use Cartesian coordinates or spherical polar coordinates.)

SOLUTION:

The region bound by the sphere and the plane called a spherical cap

The volume of the spherical cap is

$$V = \frac {\pi (R - a)}{6} (3 h ^ {2} + (R - a) ^ {2})$$

Or

$$V = \frac {\pi (R - a) ^ {2}}{3} (3 R - (R - a)) = \frac {\pi (R - a) ^ {2}}{3} (2 R + a)$$

ANSWER

$$V = \frac {\pi (R - a)}{6} (3 h ^ {2} + (R - a) ^ {2}) = \frac {\pi (R - a) ^ {2}}{3} (2 R + a)$$