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)=6 x y+2 x ; \quad 2 \leq x \leq 5,1 \leq y \leq 3

$$

49

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Okay, so we're given the function f of X y. It's equal to six x y plus two x, and we want to find the average value of this function over this region Basically the set of all points that air co ordinate pairs and r squared Such that, uh, two is between X is between two and five. And why is between one and three? This is just a little set notation, but it's basically just, uh, describing that are is a set of all points x and y that air co ordinate pairs or basically elements of R squared such that X is between these two numbers and wise between these two numbers. Okay, so if we were just sketches out quick, that's why 12345 um, 123 Basically, we sweep out my bed. 31 Okay, basically, we just sweep out this rectangular region. Okay, so let's just recall what the average value, um or basically how to calculate for average value of a function and this are too valuable function, which is a little different from or it's a little tricky to understand, but but basically the function or the average value is calculated by doing basically the double integral over a region are over function. So after, that's why d A And then we divide that by the area of our um So basically what we're doing is finding a volume and then dividing it by its area, and that should give us the average value of our function over this region. And to generalize this a little bit more, this is actually just you can rewrite this area of our as the double angle over our off the function. One D a. So similar to how, if you take the single integral from A to B of one d x, it's actually just the length of this interval, so it just be my say so. Usually single genitals are used to Copley area, but if you take the integral of the function one it scales down in dimension. It goes from area to the length of the line and similar Lou. Similarly, if you dio double integral, if you're solving for usually use, doubling it was to sell for volume. But if you just do the double integral of the function one it scales down from volume. It's the area and its area of this region are, um, for this particular problem we don't necessarily have to solve for a double integral because it's very easy to figure out what this area is. Just a rectangle. So let's just do away I'll with times Life. So the width is just three, and then length is just too. So we get an area of six, okay, and we'll just keep this in the back burner. Just remember for later, because first we gotta deal with this. The winnable of our function are functioning. Six x y plus two X Um, I was going to a D Y d at Central? Uh, no. It's no necessarily, like neither. Neither way is necessarily, like faster and dxy y Or do I. D. X just preference for now. So let's just think about how wide ranges by ranges from 13 and then X ranges from 2 to 5. Okay, now let's assault the inner integral 123 of six x wine. That's two x d. Y. Um, and now we just want to think about every every other variable. That's not wine to be constant, so we just bring six x along for the ride. Why? Quantity squared over to reverse spiral plus two x Y 123 And then we want to plug your limits in tow. Why? Because we're integrating respect. Why? We have six x three squared over two plus six X then my is what we get when you plug in one. That would just be six x over two across two X, I believe. Let me just make sure, um oh, actually, now that I'm thinking about, there's actually a better way to do this instead of doing all this nonsense. Uh, it's actually a lot simpler to do it a different way. So that took me quite a while to do. But this next way that I'm going to actually this better way. The two. It is gonna be a lot faster. Um, when they do, first is actually just fact out a two x So we have 2 to 5123 factor two x, get three y plus one d y dx, and you'll notice that since we're integrate aspecto y first, um, what we can do is think about to access a constant. Okay, so we can just bring this to x out of the STI y integral. Okay. And then the cool thing here is we can actually do is think of this interval as entire constant this this area will evaluate to a numeric value So you can think of this whole integral as a constant within this outer DX integral. So you can actually bring this entire integral outside. So one of three of three y plus one t y and then times the integral from 2 to 5 of two x dx Andi, this is a lot simpler, uh, cackling to single inter girls that are very easy to solve for rather than doing the other way. Um and this only works when you have numeric limits, because if you had ah, variable limits, then this would lead. If you have variable limits here, then you would have a you would have variables in your answer and done so that would be good. So whenever you have variables in your lemons, you always want to keep it as a iterated integral. But if you have just a New York limits, you can separate them out into single intervals. If on this supplies and usually this only works. If you have generally relate a b c d of function of why it has a function of X Ah, Deac, si y. If it's in this form, then you can actually just separated out into its two singling girls. So C d d y of f of y times a to b g of X dx. So these two would be equal on this. Just a little thing. I keep in the back my head whenever I'm doing double angles because it's very simple to do. Anyways, back to the actual problem. Uh, so now this is gonna be a lot simpler. We have three y squared over two plus y from 1 to 3. The times, um, X squared to five. Then we just plug in. We have 27/2 plus three minus what we going? Plug in 1 3/2 plus one. And then despite this by what we get 25 minus four, this is equal. Teoh 27/2 miles. 3/2 is just 24 over to on that three minus one is just plus two and then we have 21 here. This is equal to 12 plus two. 21 is equal to 14 times 21 s o. This is the newer value for our doublet nickel. But now we're trying to find the average value. So remember, we need to divide by the area of our region, which is six. So we take this 14 times 21 to buy by six. This is the average value of effort X by over our and this is just gonna be equal to if you just plug into your calculator Should spey 49. That is our final answer.

Rutgers, The State University of New Jersey