Find the volume composed by a cylinder and a sphere

Question

As the title says find the volume composed by a cylinder and a sphere

$$\left\{ \begin{array} xx^2 + y^2 \geq 9 \\ x^2 + y^2 + z^2 \leq 25 \end{array} \right. $$ I want some help in this. Thank you.

Answer 1

first get a visual idea in your head of the solid you are calculating. It is outside a cylinder but inside a sphere, so it is like a bead. The inside radius of the hole is 3, and the radius of the sphere is 5. This problem is easier to solve using cylindrical coordinates. You start by describing the solid using cylindrical coordinates (hint: you will need the z-coordinate of the intersection of the cylinder with the sphere). Then set up and evaluate the triple integral: $$V = \iiint_D r \: dr \: dz \: d\theta$$ The z-coordinates are from the bottom to the top of the hole (intersection). The r-coordinates are from 3 to the outside of the sphere (depends on z). The $\theta$ coordinates go from zero to $2\pi$. Hope this helps!