A solid has a base in the first quadrant bounded above by the curve

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

Problem

y=x^{3}

, bounded below by the

x

–axis, and bounded on the left by the

y

–axis. Cross sections perpendicular to the

x

–axis are squares. The right‐hand boundary of the base is the vertical line

x=a

, where

a>0

is to be determined. The volume of the solid is

200

cubic units. (a) Express the volume

V(a)

of the solid as a definite integral in terms of

a

. Do not evaluate the integral. (b) Evaluate the integral obtained in part (a) to obtain a closed‐form expression for

V(a)

. (c) Find the value of

a

that makes

V(a)=200

. Round your answer to the nearest thousandth.