Summary of multiple integrals

Fubini. Iterated integration on product domains.

Change of variables. $dx\,dy\to|J|\,du\,dv$; $|J|$ is the Jacobian determinant.

Polar. $x=r\cos\theta$, $y=r\sin\theta$, $dx\,dy=r\,dr\,d\theta$.

Cylindrical. $x=r\cos\theta$, $y=r\sin\theta$, $z=z$, $dV=r\,dr\,d\theta\,dz$.

Spherical. $x=r\sin\varphi\cos\theta$, $y=r\sin\varphi\sin\theta$, $z=r\cos\varphi$, $dV=r^2\sin\varphi\,dr\,d\varphi\,d\theta$.

Applications. Volumes, centroids, masses with variable density, Gaussian integrals.

Conclusion. Multiple integrals reduce to iterated integrals via Fubini, and change of variables exploits symmetries to simplify computations.