Question

Help Please!!!
Find the volume of the solid that results when the region enclosed by y=sqrt(x), y=0, and x=1 and is revolved about the line y=1

Answer 1

\[V=2\pi\iint (1-y)dxdy=2\pi\int\limits_0^1\left(\int\limits_0^{\sqrt{x}}(1-y)dy\right)dx=\]\[=2\pi\int\limits_0^1\left(x^{1/2}-\frac{x}{2}\right)dx=2\pi\left(\frac{2}{3}x^{3/2}-\frac{x^2}{4}\right)|_0^1=\]\[=2\pi\left(\frac{2}{3}-\frac{1}{4}\right)=2\pi\cdot\frac{5}{12}=\frac{5\pi}{6}\]