Consider the region in the first quadrant bounded by the –axis

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

Problem

Consider the region in the first quadrant bounded by the

y

–axis, the line

y=3

, and the graph of

x=\ln(y+1)

. A solid is formed by taking cross sections perpendicular to the

y

–axis; each cross section is a semicircle whose diameter lies entirely within the region described. The volume of the solid must be

\pi

cubic units in order to satisfy design constraints. (a) Express the radius

r(y)

of a typical semicircular cross section at height

y

in terms of

y

. Provide justification based on the geometry of the region. (b) Write, but do not evaluate, a definite integral for the volume of the solid in terms of the unknown constant

k

that scales the semicircular cross sections (so the area of one cross section is

k\,\dfrac{1}{2}\pi r(y)^{2}

). (c) Evaluate the integral you wrote in part (b) to express the volume as a function of

k

. Then determine the value of

k

that yields a total volume of

\pi

cubic units.

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