A solid has its base in the -plane in the region bounded by the -axis

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

Problem

A solid has its base in the

xy

-plane in the region bounded by the

y

-axis, the line

y=4

, and the line

x=y

. Cross sections perpendicular to the

y

-axis are rectangles whose width equals the horizontal distance between

x=0

and

x=y

, and whose height at a value of

y

h(y)=\sqrt{y}

. All dimensions are measured in inches. (a) Express the width

w(y)

of a typical cross section as a function of

y

. (b) Write, but do not evaluate, an integral expression for the volume

V

of the solid. (c) Evaluate the integral to obtain the exact volume. (d) Determine the average value of the height function

h(y)

on the interval

1 \le y \le 4

and interpret this average height in the context of the solid.