Mathematics

Problem

A solid is generated by revolving the region in the first quadrant bounded by the

y

-axis and the graphs of

x=y^{2}

for

0\le y\le 3

and

x=y+3

for

3\le y\le 5

about the

y

-axis. Which of the following expressions gives the exact volume

V

of the solid? a)

V = \pi\Bigl[\displaystyle\int_{0}^{3}y^{2}\,dy+\int_{3}^{5}(y+3)\,dy\Bigr]

V = \pi\Bigl[\displaystyle\int_{0}^{3}(y^{2})^{2}\,dy+\int_{3}^{5}(y+3)^{2}\,dy\Bigr]

V = 2\pi\Bigl[\displaystyle\int_{0}^{3}y^{2}\,dy+\int_{3}^{5}(y+3)\,dy\Bigr]

V = \pi\Bigl[\displaystyle\int_{0}^{3}(y^{2})^{2}\,dy+\int_{3}^{5}\bigl((y+3)^{2}-(y^{2})^{2}\bigr)\,dy\Bigr]