Example 6.2.1

$F\=x y$; $R$ is the finite region bounded by the graph of $y\=x \left(1-x\right)$ and the $x$-axis.

See Example 6.1.1.

Example 6.2.2

$F\=2 x^2\+3 y^2$; $R$ is the finite region bounded by the graphs of $x\=y^2$ and $y\=3-2 x$.

See Example 6.1.2.

Example 6.2.3

$F\=2 x\+3 y\+1$; $R$ is 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 6.1.3.

Example 6.2.4

$F\=3 x^2\+2 y^2\+1$; $R$ is 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 6.1.4.

Example 6.2.5

$F\=5-3 x^2-2 y^2$; $R$ is the interior of the triangle whose vertices are $\left(0\,0\right)\,\left(1\,0\right)\,\left(0\,1\right)$.

Example 6.2.6

$F\={\left(x-3\right)}^2 {\left(y-6\right)}^2\/30$; $R$ is the interior of the triangle whose vertices are $\left(1\,3\right)\,\left(7\,4\right)\,\left(5\,9\right)$. See Example 6.1.6.

Example 6.2.7

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

Example 6.2.8

$F\=\sqrt{2-x^2-4 y^2}$; $R$ is the interior of the ellipse $x^2\+4 y^2\=1$. See Example 6.1.8.

Example 6.2.9

Calculate the volume bounded by the ellipsoid $3 x^2\+5 y^2\+7 z^2\=1$.

Example 6.2.10

Find the volume that is enclosed by the surfaces $z_1\=7-x^2-y^2$ and $z_2\=2 x\+4 y-6$ and that lies inside the right cylinder whose footprint in the plane $z\=0$ is bounded by the curves $y\=x$, $y\=2-x^2$, and $x\=0$.