Let V be the volume of the three-dimensional structure bounded by the region 0<=z<=1-x^2-y^2. If V=(a/b)*pi, where a and b are positive integers, what is a and b?
since this is just a paraboloid, just use a regular triple integral, with bounds
z: 0 to 1-x^2-y^2
y: 0 to √(1-x^2)
x: 0 to 1/√2
∫[0,1/√2]∫[0,√(1-x^2)]∫[0,1-x^2-y^2] dz dy dx
∫[0,1/√2]∫[0,√(1-x^2)] 1-x^2-y^2 dy dx
...
It might be simpler to use cylindrical coordinates.
∫[0,1/√2]∫[0,2π]∫[0,1-r^2] r dz dθ dr
Since the volume of a paraboloid of radius a and height h is
v = 1/2 πa^2 h
you should come up with 1/4 π