AVERAGE VALUE OF A FUNCTION
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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 + 1 u = 1 |
When x = 2, u = 22 + 1 u = 5 |
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