다중적분의 응용(질량·질량중심)

[T] The solid $Q=\{(x,y,z)|{x}^{2}+{y}^{2}\le 9,0\le z\le 1,x\ge 0,y\ge 0\}$ has density equal to the distance to the $xy\text{-plane}.$ Use a CAS to answer the following questions.

a. Find the mass of $Q.$ b. Find the moments ${M}_{xy},{M}_{xz},\text{and}\,{M}_{yz}$ about the $xy\text{-plane,}$ $xz\text{-plane,}$ and $yz\text{-plane,}$ respectively. c. Find the center of mass of $Q.$ d. Graph $Q$ and locate its center of mass.

Consider the solid $Q=\{(x,y,z)|0\le x\le 1,0\le y\le 2,0\le z\le 3\}$ with the density function $\rho (x,y,z)=x+y+1.$

a. Find the mass of $Q.$ b. Find the moments ${M}_{xy},{M}_{xz},\text{and}\,{M}_{yz}$ about the $xy\text{-plane,}$ $xz\text{-plane,}$ and $yz\text{-plane,}$ respectively. c. Find the center of mass of $Q.$

[T] The solid $Q$ has the mass given by the triple integral $\int_{-1}^{1}\,\int_{0}^{\frac{\pi }{4}}\,\int_{0}^{1}{r}^{2}dr\,d\theta \,dz.$ Use a CAS to answer the following questions.

a. Show that the center of mass of $Q$ is located in the $xy\text{-plane.}$ b. Graph $Q$ and locate its center of mass.

The solid $Q$ is bounded by the planes $x+4y+z=8,x=0,y=0,\text{and}\,z=0.$ Its density at any point is equal to the distance to the $xz\text{-plane}.$ Find the moment of inertia ${I}_{y}$ of the solid about the $xz\text{-plane}.$

The solid $Q$ is bounded by the planes $x+y+z=3,$ $x=0,y=0,$ and $z=0.$ Its density is $\rho (x,y,z)=x+ay,$ where $a>0.$ Show that the center of mass of the solid is located in the plane $z=\frac{3}{5}$ for any value of $a.$

Let $Q$ be the solid situated outside the sphere ${x}^{2}+{y}^{2}+{z}^{2}=z$ and inside the upper hemisphere ${x}^{2}+{y}^{2}+{z}^{2}={R}^{2},$ where $R>1.$ If the density of the solid is $\rho (x,y,z)=\frac{1}{\sqrt{{x}^{2}+{y}^{2}+{z}^{2}}},$ find $R$ such that the mass of the solid is $\frac{7\pi }{2}.$

The mass of a solid $Q$ is given by $\int_{0}^{2\sqrt{2}}\,\int_{0}^{\sqrt{8-{x}^{2}}}\,\int_{\sqrt{{x}^{2}+{y}^{2}}}^{\sqrt{16-{x}^{2}-{y}^{2}}}{({x}^{2}+{y}^{2}+{z}^{2})}^{n}dz\,dy\,dx,$ where $n$ is an integer. Determine $n$ such that the mass of the solid is $\left(2-\sqrt{2}\right)\pi .$

Let $Q$ be the solid above the cone ${x}^{2}+{y}^{2}={z}^{2}$ and below the sphere ${x}^{2}+{y}^{2}+{z}^{2}-4kz=0.$ Its density is a constant $k>0.$ Find $k$ such that the center of mass of the solid is situated $7$ units from the origin.

The solid $Q=\{(x,y,z)|0\le {x}^{2}+{y}^{2}\le 16,x\ge 0,y\ge 0,0\le z\le x\}$ has the density $\rho (x,y,z)=k.$ Show that the moment ${M}_{xy}$ about the $xy\text{-plane}$ is half of the moment ${M}_{yz}$ about the $yz\text{-plane}.$

The solid $Q$ is bounded by the cylinder ${x}^{2}+{y}^{2}={a}^{2},$ the paraboloid ${b}^{2}-z={x}^{2}+{y}^{2},$ and the $xy\text{-plane,}$ where $0<a<b.$ Find the mass of the solid if its density is given by $\rho (x,y,z)=\sqrt{{x}^{2}+{y}^{2}}.$

Let $Q$ be a solid of constant density $k,$ where $k>0,$ that is located in the first octant, inside the circular cone ${x}^{2}+{y}^{2}=9{(z-1)}^{2},$ and above the plane $z=0.$ Show that the moment ${M}_{xy}$ about the $xy\text{-plane}$ is the same as the moment ${M}_{yz}$ about the $yz\text{-plane}.$

The solid $Q$ has the mass given by the triple integral $\int_{0}^{1}\,\int_{0}^{\pi /2}\,\int_{0}^{{r}^{2}}({r}^{4}+r)dz\,d\theta \,dr.$

a. Find the density of the solid in rectangular coordinates. b. Find the moment ${M}_{xy}$ about the $xy\text{-plane}.$

The solid $Q$ has the moment of inertia ${I}_{x}$ about the $yz\text{-plane}$ given by the triple integral $\int_{0}^{2}\,\int_{-\sqrt{4-{y}^{2}}}^{\sqrt{4-{y}^{2}}}\,\int_{\frac{1}{2}({x}^{2}+{y}^{2})}^{\sqrt{{x}^{2}+{y}^{2}}}({y}^{2}+{z}^{2})({x}^{2}+{y}^{2})dz\,dx\,dy.$

a. Find the density of $Q.$ b. Find the moment of inertia ${I}_{z}$ about the $xy\text{-plane.}$

The solid $Q$ has the mass given by the triple integral $\int_{0}^{\pi /4}\,\int_{0}^{2\,\sec \,\theta }\,\int_{0}^{1}({r}^{3}\cos \,\theta \,\sin \,\theta +2r)dz\,dr\,d\theta .$

a. Find the density of the solid in rectangular coordinates. b. Find the moment ${M}_{xz}$ about the $xz\text{-plane.}$

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 $(0,a).$

A solid $Q$ has a volume given by $\iint_{D}\int_{a}^{b}dz\,dA,$ where $D$ is the projection of the solid onto the $xy\text{-plane}$ and $a<b$ are real numbers, and its density does not depend on the variable $z.$ Show that its center of mass lies in the plane $z=\frac{a+b}{2}.$

Consider the solid enclosed by the cylinder ${x}^{2}+{z}^{2}={a}^{2}$ and the planes $y=b$ and $y=c,$ where $a>0$ and $b<c$ are real numbers. The density of $Q$ is given by $\rho (x,y,z)=f'(y),$ where $f$ is a differential function whose derivative is continuous on $(b,c).$ Show that if $f(b)=f(c),$ then the moment of inertia about the $xz\text{-plane}$ of $Q$ is null.

[T] The average density of a solid $Q$ is defined as ${\rho }_{ave}=\frac{1}{V(Q)}\iiint_{Q}\rho (x,y,z)dV=\frac{m}{V(Q)},$ where $V(Q)$ and $m$ are the volume and the mass of $Q,$ respectively. If the density of the unit ball centered at the origin is $\rho (x,y,z)={e}^{-{x}^{2}-{y}^{2}-{z}^{2}},$ use a CAS to find its average density. Round your answer to three decimal places.

Show that the moments of inertia ${I}_{x},{I}_{y},\text{and}\,{I}_{z}$ about the $yz\text{-plane,}$ $xz\text{-plane,}$ and $xy\text{-plane,}$ respectively, of the unit ball centered at the origin whose density is $\rho (x,y,z)={e}^{-{x}^{2}-{y}^{2}-{z}^{2}}$ are the same. Round your answer to two decimal places.

Why is it easy to use “mass” and “weight” interchangeably, even though they are different measures?