Let V be the volume of the 3-dimensional structure bounded by the paraboloid z=1−x^2−y^2, planes x=0, y=0 and z=0 and by the cylinder x^2+y^2−x=0. If V=aπ/b, where a and b are coprime positive integers, what is the value of a+b?

Answer 1

The cylinder is centered at (1/2,0) and has radius 1/2

Seems the best way to handle this one is with cylindrical coordinates (like polar coordinates, plus a z-axis):

As with polar coordinates, the alement of area is

dA = r dr dθ

so, the volume element is

dV = r dr dθ dz

We have z = 1-(x^2+y^2) = 1-r^2
x^2+y^2-x = 0 becomes
r^2 - rcosθ = 0, or
r = 1-cosθ

So,

V = ∫[0,pi/2] ∫[0,cosθ] z r dr dθ
= ∫[0,pi/2] ∫[0,cosθ] r(1-r^2) dr dθ
= ∫[0,pi/2] (1/2 cos^2(θ) - 1/3 cos^3(θ)) dθ
= 1/72 (18θ - 18sinθ + 9sin2θ - 2sin3θ) [0,pi/2]
= pi/8 - 2/9

Hmm. I don't get a*pi/b
Must have messed up. Check it out.

Answer 2

12