   Chapter 4.3, Problem 43E

Chapter
Section
Textbook Problem

Using Properties of Definite Integrals Given ∫ 2 6 f ( x ) d x = 10         and     ∫ 2 6 g ( x ) d x = − 2 evaluate ∫ 2 6 [ f ( x ) + g ( x ) ] d x ∫ 2 6 [ g ( x ) − f ( x ) ] d x ∫ 2 6 2 g ( x ) d x ∫ 2 6 3 f ( x ) d x

(a)

To determine

To calculate: The definite integral 26(f(x)+g(x))dx using the provided values 26f(x)dx=10 and 26g(x)dx=2

Explanation

Given:

The provided values are

26f(x)dx=10 and 26g(x)dx=2

And the integral to be evaluated is

26(f(x)+g(x))dx

Formula used:

If f(x) and g(x) are integrable on [a,b] then the function f(x)+g(x) are integrable on [a,b], and

ab(f(x)+g(x))dx=abf(x)dx+abg(x)dx

Calculation:

We will start with the integral to be evaluated. That is,

26(f(x)+g(x))dx

As the provided functions f(x) and g(x) are integrable on [2,6]

(b)

To determine

To calculate: The definite integral 26(g(x)f(x))dx using the provided values 26f(x)dx=10 and 26g(x)dx=2

(c)

To determine

To calculate: The definite integral 262g(x)dx using the provided values 26f(x)dx=10 and 26g(x)dx=2

(d)

To determine

To calculate: The definite integral 263f(x)dx using the provided values 26f(x)dx=10 and 26g(x)dx=2

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