A solid has as its base the planar region bounded above by the semicircle and below…

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

Problem

A solid has as its base the planar region bounded above by the semicircle

y=\sqrt{4\!-\!(x-1)^{2}}

and below by the

x

-axis. At each value of

y

in this region, the cross section perpendicular to the

y

-axis is a square. (a) Rewrite the equation of the semicircle to express

x

explicitly as a function of

y

for the portion lying to the right of the

y

-axis. (b) Using your result from part (a), express the side length of a typical square cross section as a function of

y

. (c) Formulate a definite integral that represents the volume of the solid. (d) Evaluate the integral to determine the exact volume.