Answer to Question #152132 in Calculus for Ather

Question

Find the volume of the solid inside the sphere x^2+y^2+z^2=9 and inside the x^2+y^2=1clinder

Answer 1

$x^2+y^2+z^2=9$

and

$x^2+y^2=1$

Using

$r^2 =x^2 +y^2$

$r^2 =1$

$z=\sqrt{ (9-r^2)}$

The volume of the solid in cylindrical coordinates is:

$V= \int_{0}^{2\pi} \int_{0}^{1} \int_{-\sqrt{9-r^2}}^{\sqrt{9-r^2}} rdzdrdθ$

Evaluating the first integral, we have:

$V= \int_{0}^{2\pi} \int_{0}^{1}2r \sqrt{9-r^2}drdθ$ ...

Evaluating the second integral, we have:

$V= \int_{0}^{2\pi} 18 - \frac {32\sqrt2}{3}dθ$

Evaluating in terms of the angle, we have;

$V= 18(2\pi) - \frac {32\sqrt2}{3} (2\pi)$

$V= 36\pi - \frac {64 \pi\sqrt2}{3}$