   Chapter 5.1, Problem 13E

Chapter
Section
Textbook Problem

# Sketch the region enclosed by the given curves and find its area. y = 12 − x 2 ,   y = x 2 − 6

To determine

To:

Sketch the region and find the enclosed area.

Explanation

1) Concept:

Formula:

The area A of the region bounded by the curves y=f(x), y=g(x) and the lines x=a and x=b  is

A= abfx-gxdx

fx-gx=fx-gx when fxg(x)gx-fx when gxf(x)

2) Given:

y=12-x2 and   y=x2-6

3) Calculation:

The point of intersection occurs when both the equation are equal to each other, that is,

12-x2=x2-6

18=2x2

x2=9

Using the square root property

x=3 or   x=-3

Thus, the points of intersection are at   x=-3 and   x=3. The region is sketched in the following figure.

Here, 12-x2x2-6 when   -3x3

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