Let region in the –plane be bounded above by and below by the –axis on the interval

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

Problem

Let region

S

in the

xy

–plane be bounded above by

y=\sqrt{x}

and below by the

x

–axis on the interval

0\le x\le 4

. Two solids share

S

as a common base. • Solid C has cross sections taken perpendicular to the

x

–axis that are squares. • Solid D has cross sections taken perpendicular to the

x

–axis that are rectangles whose height at each

x

equals one–half the side length of the corresponding square cross section of Solid C. Respond to each part. All work must be shown. (a) Express the side length

s(x)

of the square cross section of Solid C as a function of

x

. Hence write an algebraic expression for the cross–sectional area

A_{C}(x)

. (b) Express the cross–sectional area

A_{D}(x)

of Solid D in terms of

x

. (c) Write, but do not yet evaluate, a definite integral representing the volume

V_{C}

of Solid C, and a separate definite integral representing the volume

V_{D}

of Solid D. (d) Evaluate both integrals to obtain

V_{C}

and

V_{D}

, then compute the ratio

\dfrac{V_{D}}{V_{C}}

. Simplify your answer.

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