Mathematics

Problem

Let

k

be a positive constant. A solid has a base in the

xy

–plane bounded above by the horizontal line

y=9

and below by the parabola

y=kx^{2}

. Cross sections taken perpendicular to the

x

–axis are squares. The volume of the solid is

108

cubic units. (a) In terms of

k

, find the

x

–coordinate of the points where the curves

y=9

and

y=kx^{2}

intersect. (b) Set up, but do not evaluate, a definite integral expression for the volume of the solid in terms of

k

. (c) Determine the value of

k

that produces a volume of

108

cubic units.