Question

Use cylindrical coordinates. Find the volume of the solid that lies within both the cylinder x2 y2

Answer 1

hello your question is incomplete here is the complete question

Use cylindrical coordinates Find the volume of the solid that lies within both the cylinder x^2 + y^2=16 and the sphere x^2 + y^2 + Z^2= 81

Answer :
`(4 \pi )/(3) [729 - 65√(65) ]`

Explanation:

The given data

cylinder = x^2 + y^2 = 16

sphere = x^2 + y^2 +z^2 = 81

from the given data the solid is symmetric around the xy plane hence we will calculate half the solid volume above the plane then multiply the sesult by 2

Note : we are restricting our attention to the cylinder x^2 + y^2 = 16 and also finding the volume inside the sphere which gives bound on the z-coordinate as well

the r parameter goes from 0 to 4

ATTACHED IS THE REMAINING PART OF THE SOLUTION

showing the integration