Multiple Integrals

For a rectangular region $S$ defined by $x$ in $[a,b]$ and $y$ in $[c,d]$, the double integral of a function $f(x,y)$ in this region is given as $\int_c^d(\int_a^b f(x,y) dx) dy$.

An iterated integral is the result of applying integrals to a function of more than one variable.

Double integrals can be evaluated over the integral domain of any general shape.

When domain has a cylindrical symmetry and the function has several specific characteristics, apply the transformation to polar coordinates.

When the function to be integrated has a cylindrical symmetry, it is sensible to integrate using cylindrical coordinates.

When the function to be integrated has a spherical symmetry, change the variables into spherical coordinates and then perform integration.

For $T \subseteq R^3$, the triple integral over $T$ is written as $\iiint_T f(x,y,z)\, dx\, dy\, dz$.

One makes a change of variables to rewrite the integral in a more "comfortable" region, which can be described in simpler formulae.

Multiple integrals are used in many applications in physics and engineering.

The center of mass for a rigid body can be expressed as a triple integral.