Spherical Change of Variables

System

• $x=\rho \sin (\varphi )\cos (\theta )$
• $y=\rho \sin (\varphi )\sin (\theta )$
• $z=\rho \cos (\varphi )$

Usage

$${\iiint }_{E}f(x,y,z)dV={{\iiint }_{E^{\prime }}f(x,y,z)|}_{\text{above}}J(\rho ,\varphi )d{V}^{\prime }$$

$J={\rho }^2\sin \varphi$

Use When

• E is bounded by…
  • Spheres
  • Cones
  • Half planes

Natural Ranges

• $\rho \in [0,\infty )$
• $\varphi \in [0,\pi ]$
• $\theta \in [0,2\pi ]$

Other Equivalences

• $\sqrt{x^2+y^2+z^2}=\rho$
• $\sqrt{x^2+y^2}=\rho \sin \varphi$