Change of Variables Summary

• The double integral in polar coordinates

  • $\iint^ {}_{R}{f(x,\ y)\ dA}=\int^{\beta}_{\alpha}{\int^{b}_{a}{f\left(r\cos\theta,\ r\sin\theta\right)\ rdrd\theta}}$

AND $\iint^ {}_{D}{f(x,\ y)\ dA}=\int^{\beta}_{\alpha}{\int^{h_{2}(\theta)}_{h_{1}(\theta)}{f\left(r\cos\theta,\ r\sin\theta\right)\ rdrd\theta}}$

• Note that $dV=rdrd\theta$

• The triple integral in cylindrical coordinates

  • $\iiint^ {}_{E}{f(x,\ y,\ z)\ dV}=\int^{\beta}_{\alpha}{\int^{h_{2}(\theta)}_{h_{1}(\theta)}{\int^{u_{2}(r\cos\theta,\ r\sin\theta)}_{u_{1}(r\cos\theta,\ r\sin\theta)}{f\left(r\cos\theta,\ r\sin\theta\right)\ rdzdrd\theta}}}$

  • Note that $dA=rdzdrd\theta$

  • Typically used when you see $x^{2}+y^{2}$, when $D$ is circular, or when the integrand is easier in cylindrical coordinates

• The triple integral in spherical coordinates

  • $\iiint^ {}_{E}{f(x,\ y,\ z)\ dV}$

$=\int^{d}_{c}{\int^{\beta}_{\alpha}{\int^{g_{2}(\theta,\ \phi)}_{g_{1}(\theta,\ \phi)}{f(}}\rho\cos\theta\sin\phi,\rho\sin\theta\sin\phi,\ \rho\cos\phi)\ \rho^{2}\sin\phi\ d\rho d\theta d\phi}$

• Note that $dV=\rho^{2}\sin\phi\ d\rho d\theta d\phi$

• Typically used when you see $x^{2}+y^{2}+z^{2}$ or when $E$ involves cones/spheres

• Change of variables for double integrals (under $T:S\rightarrow R$)

  • $\iint^ {}_{R}{f(x,\ y)\ dA=\iint^ {}_{S}{f\left(g(u,\ v),h(u,\ v)\right)}}\left|\frac{\partial(x,\ y)}{\partial(u,\ v)}\right|\ dudv$, where $\frac{\partial(x,\ y)}{\partial(u,\ v)}=\left|\begin{matrix}\frac{\partial x}{\partial u} & \frac{\partial y}{\partial u}\\ \frac{\partial x}{\partial v} & \frac{\partial y}{\partial v} \end{matrix}\right|$

  • $u=f(x,\ y)$ and $v=g(x,\ y)$ are useful in finding limits of integration

  • $x=x(u,\ v)$ and $y=y(u,\ v)$ are useful for finding $\frac{\partial(x,\ y)}{\partial(u,\ v)}$

  • Substitute using whichever is most convenient