$\frac{{\int }_0^1{\int }_0^{{\sqrt{1-x^2}}_{}}\left(x^2\+y^2\right) \mathrm{dy} \mathrm{dx}}{\mathrm{\π}\/4}$ = $\frac{\mathrm{\π}\/8}{\mathrm{\π}\/4}\=\frac{1}{2}$

Example 6.4.1

Find the average value of $F\=x y$ over $R$, the finite region bounded by the graph of $y\=x \left(1-x\right)$ and the $x$-axis. See Example 6.2.1 and Example 6.1.1.

Example 6.4.2

Find the average value of $F\=2 x^2\+3 y^2$ over $R$, the finite region bounded by the graphs of $x\=y^2$ and $y\=3-2 x$. See Example 6.2.2 and Example 6.1.2.

Example 6.4.3

Find the average value of $F\=2 x\+3 y\+1$ over $R$, the region bounded by the graphs of $f\left(x\right)\=\sin \left(x\right)$ and $g\left(x\right)\=\sin \left(2 x\right)$ on $0\le x\le \mathrm{\π}$. See Example 6.2.3 and  Example 6.1.3.

Example 6.4.4

Find the average value of $F\=3 x^2\+2 y^2\+1$ over $R$, the region bounded by the graphs of $f\left(x\right)\=\arctan \left(x\+1\right)-1\/2$ and $g\left(x\right)\=\sin \left(x\right)$ on the interval $x\in \left[0\,\mathrm{\π}\/2\right]$.

See Example 6.2.4 and Example 6.1.4.

Example 6.4.5

Find the average value of $F\=2-{\left(y-1\/2\right)}^2$ over $R$, the region bounded by the graphs of 1, $\cos \left(x\right)$, and $y\=x$ on $0\le x\le 1$. See Example 6.2.7 and Example 6.1.7.

Example 6.4.6

Find the average value of $F\left(r\,\mathrm{\θ}\right)\=1\+2 r \cos \left(\mathrm{\θ}\right)\+3 r \sin \left(\mathrm{\θ}\right)$ over $R$, the interior of the loop of the 4-leaf rose $r\=\cos \left(2 \mathrm{\θ}\right)$ that straddles the positive $x$-axis. See Example 5.7.1.

Example 6.4.7

Find the average value of $F\left(r\,\mathrm{\θ}\right)\=4-3 r \cos \left(\mathrm{\θ}\right)\+5 r \sin \left(\mathrm{\θ}\right)$ over $R$, the region that is inside the cardioid $r\=1\+ \cos \left(\mathrm{\θ}\right)$ but outside the circle $r\=1$. See Example 5.7.2.

Example 6.4.8

Find the average value of $F\left(r\,\mathrm{\θ}\right)\=r^2$ over $R$, the region that is inside the circle $r\=3 \sin \left(\mathrm{\θ}\right)$ but outside the cardioid $r\=1\+\sin \left(\mathrm{\θ}\right)$. See Example 5.7.3.

Example 6.4.9

Find the average value of $F\left(r\,\mathrm{\θ}\right)\=r^2\left(2 {\cos }^2\left(\mathrm{\θ}\right)\+3 {\sin }^2\left(\mathrm{\θ}\right)\right)$ over $R$, the region that is inside the circle $r\=3 \cos \left(\mathrm{\θ}\right)$ but outside the cardioid $r\=1\+\cos \left(\mathrm{\θ}\right)$. See Example 5.7.4.

Example 6.4.10

Find the average value of $F\left(r\,\mathrm{\θ}\right)\=r\+\cos \left(\mathrm{\θ}\right)$ over $R$, the region that is inside the circle $r\=3 \cos \left(\mathrm{\θ}\right)$ but outside the limaçon $r\=2-\cos \left(\mathrm{\θ}\right)$. See Example 5.7.5.