This should look familiar!
We can now extend this to three dimensions to define a cylinder using cylindrical co-ordinates. Cylindrical co-ordinates are effective a cartesian extension of polar co-ordinates, so there is a radius, and a co-ordinate.
The translation from cylindrical to cartesian co-ordinates is as follows:
And from cartesian to cylindrical we have:
For instance, a cylinder with height 1, and radius 2 is defined as:
If we want to find the volume of a cylinder, we must 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 , as we have:
Finding the surface area of a cylinder requires us to find a parameterization in terms of two parameters. Because this cylinder 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.