## Elements of Volume and Surface Area in Spherical Coordinates

We can find a volume element in spherical coordinates by approximating a cuboid as shown.

The cuboid has sides
$dr, \; r d \theta , \; r sin \theta d \phi$
at right angles, so the volume of the cuboid
$dV \simeq r^2 sin \theta dr d \theta d \phi$
.
The approximation becomes better as
$dr, \; d \phi \; d \theta \rightarrow 0$
.
We can also approximate an element of surface area as the area of a rectangle of base and height
$r sin \theta d \phi , \; r d \theta ,$
respectively so
$dA \simeq r d \phi d \theta$
.
Again the approximation improves as
$d \phi \; dz \rightarrow 0$
.