Consider the curves and for

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

Problem

Consider the curves

y=\dfrac{x}{x+1}

and

y=\dfrac{x^{2}}{4}

for

x\ge0

. Let region

A

be enclosed by the two curves between their left‐most and right‐most points of intersection. A second region

B

is the set of all points that lie below

y=2

and above

y=\dfrac{x^{2}}{4}

, with

x

ranging from the right‐most intersection point of the two curves to

x=3

. The composite region

C

is defined as

A\cup B

. (a) Show algebraically that the curves

y=\dfrac{x}{x+1}

and

y=\dfrac{x^{2}}{4}

intersect at exactly two distinct

x

–values, and state those values. (b) Set up, without integrating, the definite integral(s) required to compute the exact area of region

A

. Include correct limits of integration and indicate which curve is the upper function on each subinterval. (c) Evaluate the integral(s) from part (b) to find the exact area of region

A

. (d) Similarly, set up and evaluate the integral expression that yields the exact area of region

B

. (e) Combine your results to find the exact area of the composite region

C

. Then determine the average height of region

C

by dividing its area by its total width along the

x

–axis. Interpret your answer.

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