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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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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 and g are continuous on [a, b], then

a b [ f ( x ) + g ( x ) ] d x = ( a b f ( x ) d x ) ( a b g ( x ) d x )

To determine

Whether the statement ab[f(x)g(x)]dx=(abf(x)dx)+(abg(x)dx) is true or false.

Explanation

Given information:

The function is ab[f(x)g(x)]dx=(abf(x)dx)+(abg(x)dx) (1)

Calculation:

Check the statement is true or false using an example as shown below:

Let assume a=0,b=2,andf(x)=g(x)=1.

Find the integral value of left hand side part in Equation (1).

Substitute 0 for a, 2 for b, 1 for f(x), and 1 for g(x) in Equation (1).

ab[f(x)g(x)]dx=02[1]dx=[x]02=20=2 (2)

Find the integral value of right hand side part in Equation (1).

Substitute 0 for a, 2 for b, 1 for f(x), and 1 for g(x) in Equation (1)

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