$m\= \int {\int }_R\mathrm{\ρ} \mathrm{dA}$ is the total mass

$M_y\= \int {\int }_R\mathrm{\ρ} x \mathrm{dA}$ is the total first moment about the $y$-axis$$

$M_x\= \int {\int }_R\mathrm{\ρ} y \mathrm{dA}$ is the total first moment about the $x$-axis$$

Example 6.5.1

Determine the coordinates of the centroid of $R$, the region bounded by $y\=\sqrt{1-x^2}$ and the $x$-axis.

Example 6.5.2

Determine the coordinates of the center of mass of a lamina in the shape of $R$ in Example 6.5.1 if its density is equal to the square of the distance from the point $\left(0\,1\/2\right)$.

Example 6.5.3

If $a\,b\,c$ are positive, and $a\>b$, determine the coordinates of the centroid of the triangle whose vertices are $\left(0\,0\right)\,\left(a\,0\right)\,\left(b\,c\right)$.

Example 6.5.4

Calculate the coordinates of the center of mass for the triangular lamina whose vertices are $\left(0 \,0\right)\,\left(0\,3\right)\,\left(4\,0\right)$, and whose density is $\mathrm{\ρ}\=1\+2 x\+ y\/2$.

Example 6.5.5

Determine the coordinates of the center of mass for a semicircular lamina whose density is equal to the distance from the center of the circle, and whose radius is 1.

Example 6.5.6

Calculate the coordinates of the center of mass of the lamina whose shape is that of the cardioid $r\=1\+\cos \left(\mathrm{\θ}\right)$ and whose density is $\mathrm{\ρ}\=\left(1\+2 x^2\+3 y^2\right)\/10$.

Example 6.5.7

Calculate the coordinates of the center of mass of the lamina whose density is $\mathrm{\ρ}\=\left(1\+2 x^2\+3 y^2\right)\/10$, and whose shape is that of $R$, the region bounded by $x\=y^2$ and

$y\=x-2$.

Example 6.5.8

Calculate the coordinates of the center of mass of the lamina whose shape is that of an isosceles right triangle with legs of length 1, and whose density is twice the square of the distance from the vertex of the right angle. Hint: Put the right angle at the origin.

Example 6.5.9

Calculate the coordinates of the centroid of $R$, the region bounded by the graphs of $x\=0$, and $x\=y-y^3$, if $y\in \left[0\,1\right]$.

Example 6.5.10

Determine the coordinates of the center of mass of the lamina whose density is $\mathrm{\ρ}\=3\+2 x\+3 y$, and whose shape is that of $R$, the region bounded by the ellipse

$x^2\+4 y^2\=1$.