Question

In words, describe when it would be advantageous to compute triple integrals to switch from rectangular coordinates to cylindrical coordinates and from rectangular coordinates to spherical coordinates. Also, describe when it would not be advantageous to switch from rectangular coordinates to cylindrical and spherical coordinates to compute a triple integral.

Answer 1

Triple integrals are used to compute the volume under a surface in 3D. For simplicity, we can resolve the rectangular coordinates into cylindrical or spherical coordinates. Let's see when it is favourable to transform the coordinates.

Rectangular to cylindrical:

When the surface involved is a paraboloid, cylinder or cone. It is easier to resolve the algebraic expression in cylindrical coordinates.

Rectangular to spherical:

When the surface involved is a sphere, cone and intersection of other surfaces.

Not suitable:

Generally, when the surface involved is too complicated or consists of intersection of different surfaces, it is easier to resolve the algebraic expression in rectangular coordinates.