Mathematics

Problem

The region bounded by

x = y

and

x = y^{2}

for

0\le y\le 2

is revolved about the vertical line

x = 3

. Which of the following expressions represents the volume

V

of the resulting solid? a)

V = \pi\displaystyle \int_{0}^{1} \!\bigl[(3-y)^{2}-(3-y^{2})^{2}\bigr] \,dy + \pi\int_{1}^{2} \!\bigl[(3-y^{2})^{2}-(3-y)^{2}\bigr] \,dy

V = \pi\displaystyle \int_{0}^{1} \!\bigl[(3-y^{2})^{2}-(3-y)^{2}\bigr] \,dy + \pi\int_{1}^{2} \!\bigl[(3-y)^{2}-(3-y^{2})^{2}\bigr] \,dy

V = \pi\displaystyle \int_{0}^{2} \!\bigl[(3-y)^{2}-(3-y^{2})^{2}\bigr] \,dy

V = \pi\displaystyle \int_{0}^{\sqrt{3}} \!\bigl[(3-y^{2})^{2}-(3-y)^{2}\bigr] \,dy + \pi\int_{\sqrt{3}}^{2} \!\bigl[(3-y)^{2}-(3-y^{2})^{2}\bigr] \,dy