Multiple Integrals
Integrating over regions, volumes, and solids in two and three dimensions.
Get the Reference SheetDouble Integrals
Evaluate iterated double integrals of a polynomial over a rectangular region.
Double Integrals Over General Regions
Evaluate double integrals over regions bounded by curves rather than straight edges — Type I and Type II regions.
Applications of Double Integrals
Mass, moments, center of mass, and moment of inertia of a lamina, via double integrals.
Double Integrals in Polar Coordinates
Convert a double integral over a disk or annulus to polar coordinates and evaluate it.
Surface Area
The surface area of z = f(x, y) over a region D, via a double integral.
Triple Integrals Over a Box
Evaluate an iterated triple integral of a polynomial over a rectangular box.
Triple Integrals in Cylindrical Coordinates
Convert a triple integral over a solid cylinder to cylindrical coordinates and evaluate it.
Triple Integrals in Spherical Coordinates
Convert a triple integral over a ball to spherical coordinates and evaluate it.
Change of Variables and the Jacobian
The general substitution rule for double integrals — the Jacobian determinant that makes any change of variables legal.