   Chapter 5, Problem 62RE ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# Let f ( x ) = { − x − 1 if − 3 ≤ x ≤ 0 − 1 − x 2 if 0 ≤ x ≤ 1 Evaluate ∫ − 3 1 f ( x )   d x by interpreting the integral as a difference of areas.

To determine

To calculate: The value of the integral function 31f(x)dx by interpreting the integral as a difference of areas.

Explanation

Given information:

The integral function is 31f(x)dx.

The function is f(x)={x1 if3x01x2 if0x1.

The region lies between x=3 and x=1.

Consider the function as [f(x)]1 and [f(x)]2 as shown below.

[f(x)]1=x1 if3x0 (1)

[f(x)]2=1x2 if0x1 (2)

To find the value of the functions within the intervals (3,0) and (0,1) as shown below.

Substitute 3 for x in Equation (1).

f(3)=(3)1=2

Find the value of the function [f(x)]1 using Equation (1).

Substitute 2 for x in Equation (1).

f(2)=(2)1=1

Find the value of the function [f(x)]1 using Equation (1).

Substitute 1 for x in Equation (1).

f(1)=(1)1=0

Find the value of the function [f(x)]1 using Equation (1).

Substitute 0 for x in Equation (1)

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