Mathematics

Problem

The region enclosed by the graphs of

y=x

and

y=x^{2}

for

0\le x\le 2

is revolved about the

x

-axis. Which of the following expressions gives the exact volume of the resulting solid? a)

\pi\displaystyle\int_{0}^{2}\bigl(x^{2}-(x^{2})^{2}\bigr)\,dx

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

\pi\left[\displaystyle\int_{0}^{1}\bigl(x^{2}-x^{4}\bigr)\,dx+\int_{1}^{2}\bigl(x^{4}-x^{2}\bigr)\,dx\right]

\pi\left[\displaystyle\int_{0}^{1}\bigl(x^{2}-x^{4}\bigr)\,dx+\int_{1}^{2}\bigl(x^{2}-x^{4}\bigr)\,dx\right]