# SOLVED: We want to find the average of f(x) over the interval [0,2] where f(x) is the function f(x) = 5x. Using the formula for average value: average = (1/(b-a)) * âˆ«[a,b] f(x) dx In this case, a =

How to find the average value of a function with integration
How to find the average value of a function with integration

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We want to find the average of f(x) over the interval [0,2] where f(x) is the function f(x) = 5x.
Using the formula for average value:
average = (1/(b-a)) * âˆ«[a,b] f(x) dx
In this case, a = 0 and b = 2, so the average value can be calculated as:
average = (1/(2-0)) * âˆ«[0,2] 5x dx
Simplifying the integral:
average = (1/2) * âˆ«[0,2] 5x dx
Now we can solve the integral:
average = (1/2) * [ (5/2) * x^2 ] evaluated from 0 to 2
average = (1/2) * ( (5/2) * 2^2 – (5/2) * 0^2 )
average = (1/2) * ( (5/2) * 4 – (5/2) * 0 )
average = (1/2) * ( (5/2) * 4 )
average = (1/2) * ( 10 )
average = 5
Therefore, the average value of f(x) over the interval [0,2] is 5.

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