# Mean of a function (1 Viewer)

#### nicoling

##### New Member
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

##### Well-Known Member
Consider this example: Suppose you have a moving object, whose velocity varies according to some velocity function $\bg_white v(t)$ between $\bg_white t = 0$ to $\bg_white t = T$. Now, what is the total distance $\bg_white D$ travelled by the object between these 2 times? The answer is obviously:
$\bg_white D=\int_0^Tv(t)dt$
Now ask yourself: what is the average velocity $\bg_white \bar{v}$? Well, the average velocity is simply the total distance travelled divided by the total time taken. Therefore:
$\bg_white \bar{v}=\frac{D}{T}=\frac{1}{T}\int_0^Tv(t)dt$
Or alternatively,
$\bg_white \int_0^Tv(t)dt=\bar{v}T$
The LHS is the area under the curve, and the RHS is the area under a rectangle with a height of $\bg_white \bar{v}$, which is the mean of the y-values.

Last edited: