# Thread: [SOLVED] Average Value in FTC?

1. ## [SOLVED] Average Value in FTC?

Hi hows it going

In the Fundamental Theorem of Calculus;

$\displaystyle \Delta F \ = \ F(b) \ - \ F(a) \ and \ \Delta x \ = \ b \ - \ a$

we can rewrite this as;

$\displaystyle \Delta F \ = \ \int_{a}^{b} f (x)\,dx$

Then if we multiply both sides by 1/Δx we get;

$\displaystyle \frac{\Delta F}{ \Delta x} \ = \ \frac{1}{b \ - \ a} \int_{a}^{b} f (x)\,dx$

This is called the Average of the function f.

What does this mean?

I was always extremely bad at any form of statistics because I didn't understand it but maybe now I'll get it. Would this be like those graphs of average rainfall throughout the year where they showed the 12 months and the rainfall in each month and you had to find the average for the year?

It just makes very little sense to me and I don't know what it's good for. When would this be useful to implement?

2. The average value is a function tells what the average y value would be over a specific range. For example the average value of $\displaystyle f(x)=x$ from $\displaystyle [0,2]$ is 1. If your wondering how they accrue the value, look at the formula. Integrating over $\displaystyle [a,b]$ gives you the area under the curve, now imagine morphing that area into a rectangle with a length of $\displaystyle b-a$. Then your height would be the average value of the function.

3. Yeah it kind of disorientated me because I was picturing weird curves in my head but I wasn't making the connection to integration. I also found a really helpful video on the topic Finding the average value of a function Video ? 5min.com .