Let be the region bounded on the left by the –axis

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

Problem

Let

R

be the region bounded on the left by the

y

–axis, bounded above by the line

y=3

, and bounded below by the curve

y=\dfrac{1}{4}x^{2}

. Two solids are built on the base

R

with cross sections taken perpendicular to the

y

–axis. • Solid C has square cross sections whose sides lie in the plane of

R

. • Solid D has right isosceles triangular cross sections whose legs lie in the plane of

R

and are perpendicular to the

y

–axis. (a) Derive an expression for the length

\ell(y)

of the base of each cross section at height

y

, expressed as a function of

y

. (b) Write, but do not evaluate, a definite integral for the volume of each solid in terms of

\ell(y)

and appropriate geometric constants. (c) Determine which solid has the greater volume and calculate the exact numerical difference

V_{C}-V_{D}

. Show all steps in your reasoning, including any algebraic simplifications of the integrals. (d) Suppose both solids are melted down and recast into identical right circular cylinders of height

1

. Determine the ratio of the radii of the cylinders obtained from Solids C and D, respectively.

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