Use polar coordinates to compute the volume of the given solid. Inside the sphere x^2 + y^2 + z^2...

Question:

Cylindrical Coordinates:

Since our region has a cylinder in it, it will be most convenient to use cylindrical coordinates, which are the usual polar coordinates with an extra dimension, {eq}z {/eq}, added in to manage depth. The relations between polar and Cartesian coordinates are

{eq}x = r \cos \theta {/eq}

{eq}y = r \sin \theta {/eq}

{eq}z = z {/eq}

{eq}r^2 = x^2+y^2 {/eq}

{eq}\theta = \tan^{-1} \frac{y}{x} {/eq}

{eq}dV = r \ dz \ dr \ d\theta {/eq}

Answer and Explanation: 1