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

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## Recommended Questions

Writing Discuss how computing a volume using an iterated double integral corresponds to the method of computing a volume by slicing (Section 6.2).

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In the following exercises, the function $f$ is given in terms of double integrals.

a. Determine the explicit form of the function $f$

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c. Find the average value of the function $f$ on $R$ .

d. Use a computer algebra system (CAS) to plot $z=f(x, y)$ and $z=f_{\text ave }$ in the same system of coordinates.

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