Let be the region in the first quadrant enclosed by the coordinate axes and the curve

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

Problem

Let

T

be the region in the first quadrant enclosed by the coordinate axes and the curve

y=\sqrt{9-x^{2}}

. Two solids are constructed on the common base

T

. • Solid M has cross sections perpendicular to the

x

–axis that are squares whose sides lie in the plane of

T

. • Solid N has cross sections perpendicular to the

x

–axis that are semicircles whose diameters lie in the plane of

T

. Solid M is made of a material with constant density

\delta_{M}

g/cm

^{3}

, and Solid N is made of a different material with constant density

\delta_{N}

g/cm

^{3}

. (a) Express each solid’s volume as a single definite integral, leaving any geometric constants explicit but unevaluated. (b) Show that the ratio of the masses of the two solids can be written in the form

\dfrac{m_{M}}{m_{N}}=k\,\dfrac{\delta_{M}}{\delta_{N}}

, where

k

is a positive real constant independent of

\delta_{M}

and

\delta_{N}

. Derive the exact value of

k

. (c) Determine the unique ratio

\dfrac{\delta_{M}}{\delta_{N}}

for which the two solids have equal mass. Simplify your result as much as possible. (d) Suppose the densities are chosen so that the two solids have equal mass. If both solids are submerged in water (density

1\text{ g/cm}^{3}

), state which solid, if either, will experience a greater net downward force due to gravity minus buoyant force. Provide a clear justification that references your work from parts (a)–(c).

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