Let T be the region in the plane bounded by the graphs of and between their points of…

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

Problem

Let T be the region in the plane bounded by the graphs of

y=x

and

y=x^{2}

between their points of intersection. The region T is revolved about the vertical line

x=3

to generate a solid. (a) Solve algebraically for the

y

-values that bound T, and express

x

as a function of

y

for each curve on that interval. (b) For a fixed value of

y

in the interval found in part (a), write complete algebraic expressions for the outer radius

R(y)

and inner radius

r(y)

of the corresponding washer produced by the revolution about

x=3

. Simplify each expression. (c) Set up, but do not yet evaluate, the definite integral that gives the volume of the solid in terms of

y

. Clearly indicate the limits of integration. (d) Evaluate the integral from part (c) to obtain the exact volume.