A region in the first quadrant is bounded on the left by the –axis

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

Problem

A region in the first quadrant is bounded on the left by the

y

–axis, below by the

x

–axis, and on the right by the vertical line

x=3

. Its upper boundary is the line

y=2x

. For each

y

–value in the interval

0\le y\le 6

, a rectangle is erected perpendicular to the

y

–axis whose base spans the horizontal width of the region at that

y

. The height of each rectangle is

k

times its base length, where

k

is a positive constant. (a) Write, but do not evaluate, a definite integral that represents the volume

V(k)

of the three–dimensional solid formed by these rectangular cross sections. (b) Given that the volume of the solid is

54

cubic units, determine the value of

k