다중적분의 응용(질량·질량중심)

Let $Q$ be the solid bounded by the $xy\text{-plane},$ the cylinder ${x}^{2}+{y}^{2}={a}^{2},$ and the plane $z=1,$ where $a>1$ is a real number. Find the moment ${M}_{xy}$ of the solid about the $xy\text{-plane}$ if its density given in cylindrical coordinates is $\rho (r,\theta ,z)=\frac{{d}^{2}f}{d{r}^{2}}(r),$ where $f$ is a differentiable function with the first and second derivatives continuous and differentiable on $(0,a).$

In Exercises 11–18, find the mass/weight of the lamina described by the region $R$ in the plane and its density function $\delta(x,y)$.

$R$ is the annulus in the first and second quadrants bounded by $x^2+y^2=9$ and $x^2+y^2=36$; $\delta(x,y) = 4$lb/ft$^2$

In Exercises 11–18, find the mass/weight of the lamina described by the region $R$ in the plane and its density function $\delta(x,y)$.

$R$ is the annulus in the first and second quadrants bounded by $x^2+y^2=9$ and $x^2+y^2=36$; $\delta(x,y) = \sqrt{x^2+y^2}$lb/ft$^2$

The average distance of a point in a plane region $D$ to a point $(a, b)$ is defined by

$$\frac{1}{A(D)}\iint _D \sqrt{(x-a)^2+(y-b)^2}\ \mathrm{d}x\,\mathrm{d}y$$

where $A(D)$ is the area of the plane region $D$. Let $D$ be the unit disk $1 \ge x^2 + y^2$. Find the average distance of a point in $D$ to the center of $D$.