Spherical coordinates. Consider the spherical coordinate system, where a point in space is represented by the coordinates $(r,,),$ with:

$x=rsincos,y=rsinsin,z=rcos$

and the volume element is $dV=r^{2}sindrdd.$

$($a$)$ Express the equation of a sphere of radius $R$ centered at the origin in spherical coordinates.

$($b$)$ Compute the integral:

$I={\displaystyle \underset{0}{\overset{2}{}}}{\displaystyle \underset{0}{\overset{}{}}}{\displaystyle \underset{0}{\overset{R}{}}}{r}^{3}sindrdd$

Interpret the result geometrically.

$($c$)$ Find the surface area of the sphere of radius $R$ by integrating over the surface:

$A={\displaystyle \underset{0}{\overset{2}{}}}{\displaystyle \underset{0}{\overset{}{}}}{R}^{2}sindd$