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 f_ave 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) = e^3x

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 + 1

u = 1

When x = 2,

u = 2² + 1

u = 5

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AVERAGE VALUE OF A FUNCTION

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