Question

Use spherical coordinates to find the volume of the region bounded by the sphere rhoρequals=8 cosine phi8cosφ and the hemisphere rhoρequals=44

Answer 1

`8\cos\varphi=4\implies\cos\varphi=\frac12\implies\varphi=\frac\pi3`

The volume is then given by


`\displaystyle\int_(\varphi=0)^(\varphi=\pi/3)\int_(\theta=0)^(\theta=2\pi)\int_(\rho=8\cos\varphi)^(\rho=4)\rho^2\sin\varphi\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi`

`=\displaystyle\frac{2\pi}3\int_(\varphi=0)^(\varphi=\pi/3)(512\cos^3\varphi-64)\sin\varphi\,\mathrm d\varphi`

`=\frac{176\pi}3`