Mean of a function (1 Viewer)

nicoling

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Hey guys,
This is actually out of the syllabus, so probably some of you guys aren't learning it but we have to do it for a research task.
We are basically studying applications of Integration; particularly- finding the mean of functions.
How would you be able to explain how the height of a rectangle with the same Area under the curve as the function is the mean of the y-values?? I am so confused, please anyone help me.
Many thanks!
 

blyatman

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Consider this example: Suppose you have a moving object, whose velocity varies according to some velocity function between to . Now, what is the total distance travelled by the object between these 2 times? The answer is obviously:

Now ask yourself: what is the average velocity ? Well, the average velocity is simply the total distance travelled divided by the total time taken. Therefore:

Or alternatively,

The LHS is the area under the curve, and the RHS is the area under a rectangle with a height of , which is the mean of the y-values.
 
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