# How do you find the volume of the region bounded by y=x^2, y=4 and x=0 and rotated about the y-axis?

Sketch the region as described in the problem and consider an element xdy extending from the y axis to the curve y=${x}^{2}$ in this region. Now using disc method form the volume integral as follows:
V= ${\int}_{0}^{4} \pi {x}^{2} \mathrm{dy}$
= ${\int}_{0}^{4} \pi y \mathrm{dy}$ , substituting ${x}^{2}$=y
=$\frac{\pi}{2} | {y}^{2} {|}_{0}^{4}$
= 8$\pi$