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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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Section
BuyFindarrow_forward

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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