Consider the region in the –plane bounded by the graph of

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

Problem

Consider the region in the

xy

–plane bounded by the graph of

y = 3\sqrt{x}

, the

x

–axis, and the vertical lines

x = 0

and

x = 3

. A solid is formed whose cross sections, taken perpendicular to the

x

–axis, are right triangles. For a slice located at position

x

, the height of the triangle is

(1+x)

times the length of its base lying in the region. (a) Show that the area of a typical cross section can be written in the form

A(x)=\dfrac{1}{2}(1+x)\,[3\sqrt{x}]^{2}

. Simplify completely. (b) Write, but do not evaluate, a definite integral expression for the volume

V

of the solid. (c) Evaluate the integral from part (b) to obtain the exact volume. (d) Let

V(t)

denote the volume of the portion of the solid extending from

x=0

to a variable upper limit

x=t

, where

0\le t\le 3

. Find

V'(t)

and interpret its meaning in the context of the solid.