Mathematics

Problem

Let

R

be the region in the first quadrant bounded by the graphs of

y = \sqrt{x}

and

y = x^{2}

. The region is revolved about the horizontal line

y = 1

to generate a solid. Which of the following definite integrals equals the volume of the solid? a)

\displaystyle \pi \int_{0}^{1} \big[(1 - \sqrt{x})^{2} - (1 - x^{2})^{2}\big] \,dx

\displaystyle \pi \int_{0}^{1} \big[(1 - \sqrt{x}) - (1 - x^{2})\big] \,dx

\displaystyle \pi \int_{0}^{1} \left[(1 - x^{2})^{2} - (1 - \sqrt{x})^{2}\right]\,dx

\displaystyle \pi \int_{0}^{1} \big[(\sqrt{x})^{2} - (x^{2})^{2}\big] \,dx