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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

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.

If f is continuous on [a, b], then

a b x f ( x ) d x = x a b f ( x ) d x

To determine

whether the given statement is true or false.

Explanation

Given information:

Show the given function as follows:

abxf(x)dx=xabf(x)dx (1)

Equation (1) is valid, if f is continuous on [a,b].

Explanation for justification of statement:

Calculation:

Consider f(x)=1 on [0,1].

Consider upper limit b as 1 and lower limit a as 0.

Substitute 1 for f(x), 0 for a, and 1 for b in Equation (1).

01x(1)dx=x01(1)dx (2)

Evaluate the part of the integral 01x(1)dx from Equation (2).

01x(1)dx=01xdx=(x22)01=12 (3)

Evaluate the part of the integral x01(1)dx from Equation (2)

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