Mathematics

Problem

Region

R

in the

xy

–plane is described as follows: for

0\le x\le 3

R

is bounded above by

y=6-x

and below by

y=x

; for

3\le x\le 6

R

is bounded above by

y=x

and below by the

x

–axis. Cross sections perpendicular to the

x

–axis are squares whose sides are the vertical distances between the bounding curves. Which of the following expressions represents the volume

V

of the solid whose base is

R

? a)

V=\displaystyle\int_{0}^{6}\bigl((6-x)-x\bigr)^2\,dx

V=\displaystyle\int_{0}^{3}\bigl((6-x)-x\bigr)^2\,dx+\int_{3}^{6}\bigl(x-0\bigr)^2\,dx

V=\displaystyle\int_{0}^{3}(6-x)^2\,dx+\int_{3}^{6}x^{2}\,dx

V=\displaystyle\int_{0}^{3}\bigl((6-x)-x\bigr)\,dx+\int_{3}^{6}\bigl(x-0\bigr)\,dx