Give an example of a region that cannot be expressed by either of the forms shown in Figure 34. (One example is the disk with a hole in the middle between the graphs of $x^{2}+y^{2}=1$ and $x^{2}+y^{2}=2$ in Figure $10 . )$

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 foreach function over the regions $R$ having the given boundaries.

## Discussion

## Video Transcript

Oh, all right. Here's another double integral over the region They were going to sketch and figure out howto the best way to set up and a girl So each of the two i over x you i d x we'Ll spend to buy the expert now but we'll figure out we may want to switch just depending on how region looks So what is the region says the region bounded by knees curso Why equals X squared? So that's just the parabola like that. Why it was excluded and y equals zero That's the X axis and then X equals two eject sickles to a vertical line. So here's our region and well, we have these nice functions of eggs. Looks like to me every fix an X value between zero and two. And where is why going? Why's going from zero up to expert? So will do I first All right. So again, excess going from zero to two and then for any fixed X between your own too. Where is why going? Why is going from zero to x squared? We have a tio I over eggs. Yeah, the Why Dex. All right and anti derivative, even to buy over X. Well, that's just going to be either the to y over eggs divided by to over Excite times except for two. So get X over too. And then the same thing The valuation from zero x squared. Okay, Miss equals zero to x over too times Yeah, to the seeing play in Ah si X square for wise too X squared over eggs Just two x king and then minus plug ins era. It's just seeing you with someone. Oh, right. So again. So let's go to Anoop Agency someone. It actually split this up into two into girls now, So I have Well, I have affect her of one half. Just sort of put that way out of the way So don't worry about it for now. Now, X, you did the two X and then minus just next look So let's see here. This guy doesn't look bad at all, So maybe we'll just go ahead and evaluate what this is. Subzero tio X to the X. Well, that's just excrement aired over to evaluated from zero to two, which is two third over to just two. So this just minus two sky over here eyes a little bit more trouble. We've got to use integration by parts. It would seem to me so. All right, we definitely want to take derivative of this X. So let's let this be Yu. And then either the two extra axle that that beat TV, OK? And so But we have This is one half, all right, So Well, it's the governors of you is X t used e x and then you see his need to x X and they is one half into the two X and you. So we have you. The city's going to be ex either the two x over two evaluated from zero to two minus the integral cured too, eh? Of e to the c vd you So another factor of a half And either the two x t x and then minus two. Okay, slowly but surely getting somewhere. So evaluate this. It too, will get to eat of the fourth over too. So that's just well, knee to the fourth evaluated zero, we get zero. This is going to give us another factor of one half someone fourth beat of the two X evaluated from zero to minus two. All right. So literally. But surely we're getting what? Heed to the fourth minus one fourth. Who? Get everyone half there times either. The fourth minus one minus two. Okay, so we have seeded fourth, minus one fourth e to the force. That's three to the fourth over four, and then we have minor. It's a plus. One fourth, minus two is Xs on, then yelps of seven over four, minus seven over four. And I I think that's good enough.

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