Let region be the disk of radius centered at the origin

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

Problem

Let region

C

be the disk

x^{2}+y^{2}\le 9

of radius

3

centered at the origin. Two solids share this circular base. • Solid R has cross sections perpendicular to the

y

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

y

–axis that are rectangles whose height is a constant

4

units at every

y

. Respond carefully to each part. (a) For a fixed

y

with

-3\le y\le 3

, express the horizontal width

w(y)

of the disk in terms of

y

. (b) Write, but do not evaluate, a definite integral for the volume

V_{R}

of Solid R. (c) Write, but do not evaluate, a definite integral for the volume

V_{S}

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

V_{R}

and

V_{S}

. Then determine the exact value of the ratio

\dfrac{V_{S}}{V_{R}}