## 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

\[ d \phi \; dz \rightarrow 0\]

.