Consider the region in the first quadrant bounded by the curves and

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

Problem

Consider the region

R

in the first quadrant bounded by the curves

x = y^2

and

x = 2y

. Region

R

is revolved about the vertical line

x = 5

, generating a solid. (a) Determine the

y

–coordinates at which

x = y^2

and

x = 2y

intersect in the first quadrant. (b) For each sub-interval in

y

found in part (a), express the outer radius and inner radius of a horizontal washer in terms of

y

. (c) Write—but do not yet evaluate—a piecewise definite integral, using

y

as the variable of integration, that gives the volume of the solid. (d) Evaluate the integral expression from part (c) to obtain the exact volume.