A solid has base region in the -plane bounded above by the horizontal line and below…

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

Problem

A solid has base region

R

in the

xy

-plane bounded above by the horizontal line

y=4

and below by the parabola

y=x^{2}

. For each

y

R

, the cross section of the solid taken perpendicular to the

y

-axis is a semicircle whose diameter lies entirely within

R

. The

x

- and

y

-coordinates are measured in inches. (a) For

0\le y\le4

, derive an expression for the diameter

d(y)

of the semicircular cross section in terms of

y

. Justify your steps algebraically. (b) Write a definite integral representing the volume

V

of the solid in cubic inches. Do not evaluate yet. (c) Evaluate the integral from part (b) to obtain

V

in exact form. (d) Convert your result from part (c) to cubic centimeters. Use the conversion

1\text{ in}=2.54\text{ cm}

and present the volume to the nearest cubic centimeter.