Posted by Saksham on Tuesday, March 12, 2013 at 9:02am.
Let V be the volume of the 3dimensional 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?

calculus  Steve, Tuesday, March 12, 2013 at 2:29pm
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 zaxis):
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) = 1r^2
x^2+y^2x = 0 becomes
r^2  rcosθ = 0, or
r = 1cosθ
So,
V = ∫[0,pi/2] ∫[0,cosθ] z r dr dθ
= ∫[0,pi/2] ∫[0,cosθ] r(1r^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.

calculus  joshua, Sunday, March 17, 2013 at 12:34pm
12
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