By using cyclindrical coordinates write a triple integral (don't evaluate) to find the volume of the solid bounded by z = 8- x^2- y^2 and z = x^2 + y^2.

Answer 1

If z <= 8 - x^2 - y^2 and z >= x^2 +y^2,

8 - x^2 - y^2 >= x^2 +y^2 or

2x^2 + 2y^2 <= 8 or

x^2 + y^2 <= 4

Thus, when we convert to cylindrical coordinates, r <= 2 and 0 <= θ < 2π

As x^2 +y^2 <= z <= 8 - x^2 - y^2, r^2 <= z <= 8 - r^2

When we convert from rectangular to cylindrical coordinates, we use r dr dθ = dx dy

Then, using I[a,b] for the integral from a to b, we have

I[0,2] I[r^2, 8 - r^2] I[0, 2π] r dθ dz dr