A 3-dimensional structure is obtained from rotating the parabola y=x^2 about the y-axis. Each second, 2π units^3 of water is being poured into the structure from the top. When 8π units^3 of water has been poured in the structure, the instantaneous change in water height level is ab, where a and b are coprime positive integers. What is the value of a+b?

Answer 1

the volume of water when y=k is
∫[0,k] πx^2 dy
= ∫[0,k] πy dy
= π/2 k^2

So, at depth y,

v = π/2 y^2
when v=8π, y=4

dv/dt = πy dy/dt
2π = π(4) dy/dt
dy/dt = 1/2

You sure you want ab, and not a/b?
a+b = 1+2 = 3

Answer 2

Yes sorry, it's a/b.