   Chapter 5.1, Problem 11E

Chapter
Section
Textbook Problem

# Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle and label its height and width. Then find the area of the region. x = 1 − y 2 ,   x = y 2 − 1

To determine

To sketch:

The region enclosed by the given curve and find the area of the region.

Explanation

1) Concept:

i. The intersection of the two curves is obtained by solving the simultaneous equation of the curves.

ii. Identify the top and the bottom boundaries of region.

Formula-

The area of the typical rectangle is

xR-xLy, where, RT= is the right boundary curve and xL is the left boundary curve.

The total area is

A= limni=1nxR-xLy = abxR-xLdy

2) Given:

x=y2-1 and x=1-y2

3) Calculation:

The given curves are x=y2-1 and x=1-y2

First find the intersection of these two curve by solving the simultaneous equation:

y2-1=1-y2

y2-1-1+y2=0

that is,

2y2-2=0

2y2-1=0

y2-1=0

Simplifying,

y+1y-1=0

y=-1 or y=1

Therefore, the points of intersection of two curves are 0, -1, and (0, 1,)

The graph of the region enclosed by the given curves is

Find the area by integrating with respect to y(since, we found the left curve and the right curve from the graph and the restriction on y is calculated above, so it is easy to find area by integrating with the respective  y)

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