A solid has a base region in the -plane bounded above by the graph of

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

Problem

A solid has a base region

R

in the

xy

-plane bounded above by the graph of

y=f(x)

, below by the

x

-axis, and between

x=a

and

x=b

where

f

is positive and continuous. At each

x

[a,b]

, the cross section of the solid perpendicular to the

x

-axis is a rectangle whose width equals the

y

-coordinate of the base at that

x

and whose height is given by a differentiable function

h(x)>0

. Units for

x

and

y

are meters. Throughout this question, assume all functions are integrable on the stated intervals. (a) Express the area

A(x)

of the rectangular cross section at position

x

in terms of

f(x)

and

h(x)

. (b) Write a definite integral that represents the volume

V

of the solid and clearly identify the integrand and limits of integration. (c) Explain, in complete sentences, why the integral you wrote in part (b) yields volume. In your explanation, discuss the physical units of the integrand and of the differential element. (d) Suppose

h(x)=2f(x)

. Describe qualitatively how doubling the height relative to the base affects the value of the volume integral without performing any computation.

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