# Volume between two hemispheres

edited April 28

Find the mass of the solid with boundary $z\leq0$ and $1 \leq x^2+y^2+z^2 \leq 4$ and with density
$$\frac{1}{\sqrt{x^2+y^2+z^2}}.$$

• edited April 30

It is easiest to evaluate this as a polar triple integral.

$$x^2+y^2+z^2=ρ^2$$

Therefore:

Density =1/sqrt(ρ^2) and 1≤ρ≤2

Because z≤0: 0≤φ≤π/2

The volume is a hemisphere so we integrate a full rotation: 0≤θ≤2π

So the triple integral is:

$$M=\int_0^{2π} \int_0^\frac{π}{2} \int_1^2 \frac{1}{ρ} ρ^2\,sinφ\,dρdφdθ.$$

$$M=\int_0^{2π} \int_0^\frac{π}{2} \int_1^2 ρ\,sinφ\,dρdφdθ.$$

$$M=\int_0^{2π} \int_0^\frac{π}{2} \frac{ρ^2}{2}\,sinφ\,dφdθ\Big|_1^2$$

$$M=\int_0^{2π} \int_0^\frac{π}{2} \frac{3}{2}\,sinφ\,dφdθ.$$

$$M=\int_0^{2π} \frac{3}{2}\,dθ.$$

$$M=3π$$