Let be the region bounded by the graphs of and for

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

Problem

Let

U

be the region bounded by the graphs of

y=x\arctan x

and

y=x^{3}

for

x\ge 0

. The region is revolved about the horizontal line

y=-2

, producing a solid of revolution. (a) Show that the curves

y=x\arctan x

and

y=x^{3}

intersect at

x=0

and at exactly one other positive value

x=B

. Justify that no additional positive intersections exist. (b) Use a numerical method to approximate

B

to four decimal places. (c) Write a definite integral that represents the volume of the solid obtained by revolving

U

about

y=-2

. (d) Evaluate the integral from part (c) numerically to three decimal places.