# AVERAGE VALUE OF A FUNCTION

Let f(x) be continuous over the interval [a, b]. Then, the average vaklue of the function f(x) or fave on [a, b] is given by

Example 1 :

Find the average value of f(x) over the interval [0, 5], where

f(x) = x + 1

Solution :

Average value of f(x) over the interval [0, 5] :

Example 2 :

Find the average value of f(x) over the interval [0, 3], where

f(x) = 6 - 2x

Solution :

Average value of f(x) over the interval [0, 3] :

Example 3 :

Find the average value of f(x) over the interval [-1, 1], where

f(x) = x2

Solution :

Average value of f(x) over the interval [-1, 1] :

In a function f(x), if

f(-x) = f(x) ----> f(x) is an even function

f(-x) = -f(x) ----> f(x) is an odd function

Here, f(x) = x2.

f(-x) = (-x)2 = x2 = f(x)

So, f(x) is an even function

Then,

Example 4 :

Find the average value of f(x) over the interval [-1, 1], where

f(x) = x5

Solution :

Average value of f(x) over the interval [-1, 1] :

Here, f(x) = x5.

f(-x) = (-x)5 = -x5 = -f(x)

So, f(x) is an odd function

Then,

Example 5 :

Find the average value of f(x) over the interval [0, 2π], where

f(x) = sinx

Solution :

Average value of f(x) over the interval [0, 2π] :

Example 6 :

Find the average value of f(x) over the interval [0, 2π], where

f(x) = cosx

Solution :

Average value of f(x) over the interval [0, 2π] :

Example 7 :

Find the average value of f(x) over the interval [0, 1], where

f(x) = e3x

Solution :

Average value of f(x) over the interval [0, 1] :

Example 8 :

Find the average value of f(x) over the interval [0, 2], where

Solution :

Average value of f(x) over the interval [0, 2] :

Let u = x2 + 1.

ᵈᵘ⁄dₓ = 2x

du = 2xdx

 When x = 0,u = 0 + 1u = 1 When x = 2,u = 22 + 1u = 5

Kindly mail your feedback to v4formath@gmail.com

## Recent Articles

1. ### Simplifying Algebraic Expressions with Fractional Coefficients

May 17, 24 08:12 AM

Simplifying Algebraic Expressions with Fractional Coefficients

2. ### The Mean Value Theorem Worksheet

May 14, 24 08:53 AM

The Mean Value Theorem Worksheet