Let be the region in the –plane bounded by the parabola and the line for

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

Problem

Let

R

be the region in the

xy

–plane bounded by the parabola

y=x^{2}

and the line

y=2x

for

x\ge 0

. The region

R

is revolved about the vertical line

x=-1

to generate a solid. Answer the following. (a) Determine the exact

x

–coordinates of the intersection points of

y=x^{2}

and

y=2x

. (b) Write, but do not evaluate, an integral expression for the volume of the solid generated by revolving

R

about

x=-1

. Use

x

as the variable of integration. (c) Evaluate the integral in part (b) to obtain the exact volume of the solid.