Volume between two hemispheres

edited April 28 in Last questions

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

Comments

  • 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:

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

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

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

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

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

    $$M=3π$$

    mark
Sign In or Register to comment.