Multiple Integrals

Multiple integrals extend single-variable integration to functions of two or three variables, computing quantities over regions in the plane and in space. This topic covers double and triple integrals in Cartesian and polar coordinates, with applications to area, volume, mass, and center of mass.

Multiple integrals are used in physics for computing mass and moments of inertia, in probability for joint distributions, and in engineering for analyzing stress and fluid flow over regions.

Practice Tips

• 1Sketch the region of integration before setting up limits; this prevents errors in choosing the inner and outer bounds.
• 2Switch to polar coordinates (r, theta) when the region or integrand involves x^2 + y^2, and remember to include the Jacobian factor r.
• 3For triple integrals, consider cylindrical or spherical coordinates when the region has axial or spherical symmetry to simplify the computation significantly.