Spherical coordinates

When to choose this method

• The region depends on x²+y²+z² or distance from the origin.
• Fix the convention: here φ is measured from the positive z axis and θ around that axis.

Before calculating

1. Describe radius, polar angle and azimuth before integrating.
2. Transform the function and multiply by ρ² sinφ. In the editor the radius is named r.

A complete example, step by step

1. The region is a full ball of radius 2. The polar angle spans π, not 2π.

   $$0\le\rho\le2,\quad0\le\phi\le\pi,\quad0\le\theta\le2\pi$$

2. The sum of squares becomes ρ².

   $$x=\rho\sin\phi\cos\theta,\quad y=\rho\sin\phi\sin\theta,\quad z=\rho\cos\phi$$

3. The function ρ² and Jacobian ρ² sinφ produce ρ⁴ sinφ.

   $$I=\int_0^{2\pi}\int_0^\pi\int_0^2\rho^4\sin\phi\,d\rho\,d\phi\,d\theta$$

4. Integrate the radial power before touching the angles.

   $$\int_0^2\rho^4\,d\rho=\frac{32}{5}$$

5. The polar integral equals 2; the primitive minus sign matters.

   $$\int_0^\pi\sin\phi\,d\phi=[-\cos\phi]_0^\pi=2$$

6. Multiply by the azimuthal integral 2π.

   $$I=\frac{32}{5}\cdot2\cdot2\pi=\frac{128\pi}{5}$$

Now try it yourself

Solve on paper before opening the hint or answer. These are self-assessment activities; to check a typed answer, open Learn.

Connect with other methods