Use polar coordinates to find the volume of the given solid

Use polar coordinates to find the volume of the given solid. Above the cone z = squareroot x^2 + y^2 and below the sphere x^2 + y^2 + z^2 = 49

Solution

Solution :

Using cylindrical coordinates (with z playing the usual role of y).
z = (x² + y²) ==> z = r
x² + y² + z² = 49 ==> z = (49 - r²), since we want z > 0 (sketch the region).

Curve of intersection: (49 - r²) = r
==> 49 - r² = r²
==> r = 7/2, a circle.

So, the volume 1 dV equals
( = 0 to 2) (r = 0 to 7/2) (y = r to (49 - r²)) 1 * (r dz dr d)
= 2 (r = 0 to 7/2) r [(49 - r²) - r] dr
= (r = 0 to 7/2) [2r(49 - r²)^(1/2) - 2r²] dr
= (343*)/3)*(2-2))