The region in the first quadrant is bounded by the curves ,

MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration

Problem

The region

R

in the first quadrant is bounded by the curves

y = x

y = x^2

, and the vertical line

x = 2

. Region

R

is revolved about the horizontal line

y = 3

, producing a solid

S

. (a) Determine the

x

–coordinate at which the graphs of

y = x

and

y = x^2

intersect on the interval

0 \le x \le 2

. (b) For each sub-interval determined in part (a), state the outer radius and the inner radius of a representative washer in terms of

x

. (c) Write—but do not yet evaluate—a piecewise definite integral expression for the volume of

S

. (d) Evaluate the integral expression obtained in part (c) to find the exact volume of

S