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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  
\[ d \phi \; dz \rightarrow 0\]
.