Find the volume of the solid whose base is the region enclosed by y=x^2 and y=2, and the cross sections perpendicular to the y-axis are squares.

Answer 1

For some value of y, the base of the square is 2x, so its area is 4x^2 = 4y.

So, add up all those thin square sheets of thickness dy, and you get

v = ∫[0,2] 4y dy = 8