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The idea of the average value of a function, discussed earlier for functions of the form $y=f(x),$ can be extended to functions of more than one independent variable. For a function $z=f(x, y),$ the average value of $f$ over a region $R$ is defined as

$$\frac{1}{A} \iint_{R} f(x, y) d x d y$$

where $A$ is the area of the region $R .$ Find the average value for each function over the regions $R$ having the given boundaries.

$$

f(x, y)=x^{2}+y^{2} ; \quad 0 \leq x \leq 2,0 \leq y \leq 3

$$

$\frac{13}{3}$

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Campbell University

Harvey Mudd College

University of Nottingham

Boston College

All right, So we're given function affects why it's equal to x squared plus y squared. And we want to find the average value of this function over the region. Specify this flypast set the set of all points X fly that are corden pairs or basically elements of R squared such that excess between zero and two and wise between zero and three. Okay, this is just a little set notation just to describe all the points that are within our region are. And let's just draw this out really quick as you can clarity gas in your head is gonna look like a rectangular region. You, uh, smarty pants out there, This rectangular region right here. And, um next, what we wanna do is recall what the average value equation is. Basically, it's just the double integral of region are of her function f x y d a. Divided by the area of our which in this case is just the area of a rectangle. But just to generalize, what this area are would be if it wasn't a, uh, normal geometric shape that we could kept in the area for easily. It would just be actually just the double integral over our of one d A. So if you recall from single variable couch not sure if you're a teacher ever taught you. But if you take the single angle of the function one DX, it's actually just going to result in the length of the interval. And you can verify this by after just doing calculating the integral just explicitly you. It has to be might be my say And this is basically just the length of the interval. So usually single intervals were used to find the area of a, uh, area between, um, dysfunction and the, uh, the X axis. But if we take the interval of the function one it gives us the length of the line are basically ones I mentioned lower, Um it goes from an area the area of a, you know, space to the length of the line. Um, and similarly, if you for double intervals, Dublin girls are usually used to find volume, and if you use double angles for the function one, it scales down from Volume two area and specifically is the area of region arm. In this case, we don't necessarily have to use this double integral to find the area because it's a a Z I mentioned before. It's a regular geometric shape that we can calculate the area for, But just keep this in the back of your head. This is actually the more generalized version of the average value equation. Okay, so let's calculate for the top integral first. So if the X Y is X squared plus twice where I didn't just do a d y dx integral. Just my preference, um so are wise. Range from 0 to 3, and our exes range from 0 to 2. Um, it solving for this is pretty simple. It's just stupid, the inner integral. First, this could be equal to X squared. Why? Because that's where it is just a constant in this case. Plus fly to third over, three sort of three looking and three. We get three X squared. Um, we plug into Why? Because we're integrated spectacle I and then plus three to third over three is just nine. And then what will you get when you plug in? Zero should just be zero if you work out in your head. Um, zero times X squared to zero, and then 0/3 deserve. Okay, so now we have the plug this into the outer Integral said that grow from 0 to 2 of three x squared plus nine DX. Something for this is pretty simple. Just actually third plus nine x evaluate from 0 to 2. Plug into. First we get eight plus 18 minus. Will we get one implement zero, which is just zero. This results in 26 and then this is basically the, um, top integral. But we knew we need so now divided by the area of our region. Art, which was, actually, we didn't caliphate for it, but it's just two times three times with just six. So you just need to buy by sex. So average value of Ethel bags. Why over region are it's just 26 to buy basics, which is just 13/3, and that is our final answer.

Rutgers, The State University of New Jersey