ABSTRACT
In this chapter we review integrals which are defined in multiple dimensions, and how to evaluate them by hand and using Python. We present several applications of multiple integrals in various areas of science and engineering, and show how the use of polar and spherical coordinates can simplify the evaluation of complex integrals. Specific examples are considered from undergraduate courses in Mechanics (e.g. evaluation of the moment of inertia tensor), in Electromagnetism (e.g. Coulomb forces and potentials for various configurations), and also in gravitational forces and potentials. Multiple integrals are evaluated both symbolically using SymPy, and numerically using SciPy libraries. The Jacobian determinant is introduced, since it is commonly encountered in the method of changing variables in multiple integrals.
