The base of a solid lies in the -plane

MATHEMATICS · Subjective · AP · Calculus AB · Applications of Integration · Applications of Integration

Problem

The base of a solid lies in the

xy

-plane. For

-1 \le x \le 0

, the region is bounded above by

y = x^{2} + 1

and below by the

x

-axis; for

0 \le x \le 2

, it is bounded above by

y = 2 - x

and below by the

x

-axis. Cross sections perpendicular to the

x

-axis are rectangles: in the interval

-1 \le x \le 0

the height of each rectangle is twice the length of its base, while in

0 \le x \le 2

the cross sections are squares. Determine the exact volume of the solid.