Multiple integrals

Double integrals over rectangles

\( \textnormal{The double integral of } g \textnormal{ over the rectangle } R \textnormal{ is } \) \begin{equation} \iint_R g(x,y) \, dA = \lim_{ p, \, q \to \infty} \sum_{w = 1}^p \sum_{l = 1}^q f(x_{wl}^*, y_{wl}^*) \Delta A \tag{1.1} \end{equation} \( \textnormal{if the limit exists.} \)

Double integrals over general regions

Double integrals in polar coordinates

Applications of double integrals

Surface area

Triple integrals

Triple integrals in cylindrical coordinates

Triple integrals in spherical coordinates

Change of variables in multiple integrals