   Chapter 5.1, Problem 18E

Chapter
Section
Textbook Problem

# Sketch the region enclosed by the given curves and find its area. y = x − 1 ,   x − y = 1

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=x-1 and   x-y=1

3) Calculation:

Write both the equations as  y in the terms of   x.

y=x-1

y=x-1

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

x-1=x-1

Squaring both the sides,

x-1=x-12

x-1=x2-2x+1

x2-3x+2=0

x-1x-2=0

x=1 and   x=2

Thus, the points of intersection are at x=1 and   x=2. The region is sketched in the following figure.

Here,x-1x-1 when   1x2

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