Similar to cylinders, a sphere is an extension of polar co-ordinates, except this time it is a polar extension as opposed to a cartesian extension. This means that spherical co-ordinates have another angle .
The translation from spherical to cartesian co-ordinates is as follows:
And from cartesian to cylindrical we have:
For instance, a sphere with radius 2 is defined as:
If we want to find the volume of a sphere, we must again use a triple integral as the input space has three parameters.
Note also that we need to apply a density correction which is the determinant of the Jacobian matrix for each component. In the case of a cylinder, that is .
Finding the surface area of a cylinder requires us to find a parameterization in terms of two parameters. Because this sphere has a constant radius, we can use a parameterization that keeps constant.
From there we can use the same approach to find the length of the normal vector at every point.
So from those points, we can find the surface area.