Find the volume formed by rotating about the y-axis the region enclosed by: x=4y and y^3=x with y>=0

Answer 1

y = x/4 and y = x^1/3; x/4 = x^1/3; x³/64 = x; x = 0; x = 8.

Volume v = 2π∫₀⁸x(- x/4 + x^1/3)dx = 2π(- x³/12 + 3/7x^7/3)₀⁸ = 76.6

Answer 2

Find the intersection point of the two equations (0,0) and (8,2) to determine the limits of integration y=0 to y=2. Calculate the volume of the cone by taking the integral from 0 to 2 of π(4y)²dy, then subtract the volume of the solid by taking the integral from 0 to 2 of π(y³)²dy.