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.

For example two lines are $x+y=2$ and $x+y=4$

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Harvey Mudd College

University of Michigan - Ann Arbor

Okay, So really compare a region they can't be expressed. I just kind of is the difference you are The region bounded between either two functions of X or pounded between two functions of why and one example that they give. So here's our first example. Is this annual ous So X squared? That's why squared We want to be inside a circle of radius two we're renting and then outside Circle Radius one. So here's circle raise to er to an outside circle. Raise one on the inside. Let me see that this is not bounded between two functions of ice and destruction of why Now it is bounded, huh? It is the region bounded between two functions of kind of the angle state. So if we look in terms of polar coordinates, this is R squared between two and one. So you know it's bounded between these two functions. Far. So another example. Well, I mean, we could do another Angelus that we could do something a little bit different, too. We could look at, say, the region, how founded between why equals X cute. Why Eagle's eggs and then say X nickel's closer, minus one. So this would be another example. So we plot this. We have excuse you. No, we have y equals X and Dan years X equals minus one. X equals one saw region like something like this. And you see that? No, this region is not the region bounded between two functions of I mean it is the region bounded between two functions of X, but the functions actually switched signs that we're gonna have to Just like you know, when we do this integral and Anya List will probably have to do like the top half. And in the bottom half here, we're gonna have to do the region between negative one to zero and in the region from zero one, because thes dysfunctions were actually switching, which is bigger rights over here, X cubed is bigger. So why's going from next X cube butt over here? Why is going from execute two x

Georgia Southern University