5.6 Calculating centers of mass and moments of inertia  (Page 6/8)

$R$ is the triangular region with vertices $\left(0,0\right),\left(0,3\right),$ and $\left(6,0\right);\rho \left(x,y\right)=xy.$

$R$ is the triangular region with vertices $\left(0,0\right),\left(1,1\right),$ $\left(0,5\right);\rho \left(x,y\right)=x+y.$

$R$ is the rectangular region with vertices $\left(0,0\right),\left(0,3\right),\left(6,3\right),$ and $\left(6,0\right);$ $\rho \left(x,y\right)=\sqrt{xy}.$

$R$ is the rectangular region with vertices $\left(0,1\right),\left(0,3\right),\left(3,3\right),$ and $\left(3,1\right);$ $\rho \left(x,y\right)=x^{2}y.$

$R$ is the trapezoidal region determined by the lines $y=-\frac{1}{4}x+\frac{5}{2},y=0,y=2,$ and $x=0;$ $\rho \left(x,y\right)=3xy.$

$R$ is the trapezoidal region determined by the lines $y=0,y=1,y=x,$ and $y=\text{−}x+3;\rho \left(x,y\right)=2x+y.$

$R$ is the disk of radius $2$ centered at $\left(1,2\right);$ $\rho \left(x,y\right)=x^2+y^2-2x-4y+5.$

$R$ is the unit disk; $\rho \left(x,y\right)=3x^4+6x^2{y}^2+3y^4.$

$R$ is the region enclosed by the ellipse $x^2+4y^2=1;\rho \left(x,y\right)=1.$

$R=\left\{\left(x,y\right)|9x^2+y^2\le 1,x\ge 0,y\ge 0\right\};$ $\rho \left(x,y\right)=\sqrt{9x^2+y^2}.$

$R$ is the region bounded by $y=x,y=\text{−}x,y=x+2,y=\text{−}x+2;$ $\rho \left(x,y\right)=1.$

$R$ is the region bounded by $y=\frac{1}{x},y=\frac{2}{x},y=1,$ and $y=2;\rho \left(x,y\right)=4\left(x+y\right).$

[T] $R$ is the triangular region with vertices $\left(0,0\right),\left(0,3\right),$ and $\left(6,0\right);\rho \left(x,y\right)=xy.$

[T] $R$ is the triangular region with vertices $\left(0,0\right),\left(1,1\right),\text{and}\left(0,5\right);\rho \left(x,y\right)=x+y.$

[T] $R$ is the rectangular region with vertices $\left(0,0\right),\left(0,3\right),\left(6,3\right),\text{and}\left(6,0\right);$ $\rho \left(x,y\right)=\sqrt{xy}.$

[T] $R$ is the rectangular region with vertices $\left(0,1\right),\left(0,3\right),\left(3,3\right),\text{and}\left(3,1\right);$ $\rho \left(x,y\right)=x^{2}y.$

[T] $R$ is the trapezoidal region determined by the lines $y=-\frac{1}{4}x+\frac{5}{2},y=0,$ $y=2,\text{and}x=0;$ $\rho \left(x,y\right)=3xy.$

[T] $R$ is the trapezoidal region determined by the lines $y=0,y=1,y=x,$ and $y=\text{−}x+3;\rho \left(x,y\right)=2x+y.$

[T] $R$ is the disk of radius $2$ centered at $\left(1,2\right);$ $\rho \left(x,y\right)=x^2+y^2-2x-4y+5.$

[T] $R$ is the unit disk; $\rho \left(x,y\right)=3x^4+6x^2{y}^2+3y^4.$

[T] $R$ is the region enclosed by the ellipse $x^2+4y^2=1;\rho \left(x,y\right)=1.$

[T] $R=\left\{\left(x,y\right)|9x^2+y^2\le 1,x\ge 0,y\ge 0\right\};$ $\rho \left(x,y\right)=\sqrt{9x^2+y^2}.$

[T] $R$ is the region bounded by $y=x,y=\text{−}x,y=x+2,$ and $y=\text{−}x+2;$ $\rho \left(x,y\right)=1.$

[T] $R$ is the region bounded by $y=\frac{1}{x},$ $y=\frac{2}{x},y=1,\text{and}y=2;$ $\rho \left(x,y\right)=4\left(x+y\right).$

$R$ is the triangular region with vertices $\left(0,0\right),\left(0,3\right),$ and $\left(6,0\right);\rho \left(x,y\right)=xy.$

$R$ is the triangular region with vertices $\left(0,0\right),\left(1,1\right),$ and $\left(0,5\right);\rho \left(x,y\right)=x+y.$

$R$ is the rectangular region with vertices $\left(0,0\right),\left(0,3\right),\left(6,3\right),$ and $\left(6,0\right);\rho \left(x,y\right)=\sqrt{xy}.$

$R$ is the rectangular region with vertices $\left(0,1\right),\left(0,3\right),\left(3,3\right),$ and $\left(3,1\right);\rho \left(x,y\right)=x^{2}y.$

$R$ is the trapezoidal region determined by the lines $y=-\frac{1}{4}x+\frac{5}{2},y=0,y=2,$ and $x=0;\rho \left(x,y\right)=3xy.$