A region in the -plane is bounded above by the parabola and below by the -axis

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

Problem

A region

R

in the

xy

-plane is bounded above by the parabola

y=4-x^{2}

and below by the

x

-axis. A solid is constructed so that every cross section perpendicular to the

y

-axis is a semicircle whose diameter lies in

R

. Students often elect to integrate with respect to

x

for problems involving areas under curves. In the context of this solid, integrating with respect to

y

is more natural. (a) Describe, without performing any integration, how the diameter of a representative semicircular cross section can be expressed as a function of

y

. Your answer should reference specific features of the region

R

. (b) Explain why integrating with respect to

y

directly yields the volume of the solid, whereas an approach integrating with respect to

x

would require partitioning the base region into separate subregions or using multiple integrals. Support your explanation with clear geometric or algebraic reasoning.

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