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).
Evaluate the double integral by first identifying it as the volume of a solid.
$\iint_{R}(4-2 y) d A, \quad R=[0,1] \times[0,1]$
Evaluate the double integral by first identifying it as the volume of a solid.
$ \iint_R (2x + 1)\ dA $, $ R = \{(x, y) \mid 0 \le x \le 2, 0 \le y \le 4 \} $
Evaluate the double integral by first identifying it as the volume of a solid.
$ \iint_R (4 - 2y)\ dA $, $ R = [0, 1] \times [0, 1] $
Evaluate the double integral by first identifying it as the volume of a solid.
$ \iint_R \sqrt{2}\ dA $, $ R = \{(x, y) \mid 2 \le x \le 6, -1 \le y \le 5 \} $
In the following exercises, the function $f$ is given in terms of double integrals.
a. Determine the explicit form of the function $f$
b. Find the volume of the solid under the surface $z=f(x, y)$ and above the region $R$ .
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.
$f(x, y)=\int_{0}^{y} \int_{0}^{x}(x s+y t) d s d t$ where $(x, y) \in R=[0,1] \times[0,1]$
Use double integration to find the volume of the solid.
(Graph cannot copy)
Use double integration to find the volume of the solid.
(Graph cannot copy)
Draw a solid whose volume is given by the double integral $\int_{0}^{6} \int_{1}^{2} 10 d y d x .$ Then evaluate the integral using geometry.
Interpret the volume integral in the Divergence Theorem.