   Chapter 5.1, Problem 59E

Chapter
Section
Textbook Problem

# Find the values of c such that the area of the region bounded by the parabolas y = x 2 − c 2 and y = c 2 − x 2 is 576.

To determine

To find:

The value of c such that the area of the region bounded by the parabolas y=x2-c2 and y=c2-x2 is 576

Explanation

1) Concept:

Put the two functions equal to each other, and find the point of intersection. These points are the integral limits.

2) Given:

The area of the region bounded by the parabolas y=x2-c2 and y=c2-x2 is 576.

3) Calculation:

To find the values of c such that the area of the region bounded by the parabolas y=x2-c2 and y=c2-x2 is 576:

Put the two functions equal to each other, and find the point of intersection.

Given that

y=x2-c2, y=c2-x2

Equate the two functions.

x2-c2=c2-x2

By simplification,

2x2=2c2

By square root,

x=±c

Area is the integral of the difference of two functions.

Area =-cc(c2-x2-x2+c2)dx

Since the functions are even,

-ccf(x)dx=20cf(x)dx

Thus,

Area =20c(2c2-2x2)dx

Solve integral

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