Mass and Center of Mass for a Lamina with Variable Density and Given Points

[harpazo]

Find the mass and center of mass of the lamina for the indicated density.

R: (0, 0), (a, 0), (0, b), (a, b)

ρ = k

I know the formula to find the mass and center of mass.

My problem is twofold:

1. Finding the limits of integration for the inner and outer integrals considering the fact that the given points include letters a and b. How on Earth am I to graph the given points to help me reach my goal here?

2. I am confused about k. Above we see rho = k. Is k the density function?

 

[Ackbach]

Find the mass and center of mass of the lamina for the indicated density.

R: (0, 0), (a, 0), (0, b), (a, b)

ρ = k

I know the formula to find the mass and center of mass.

My problem is twofold:

1. Finding the limits of integration for the inner and outer integrals considering the fact that the given points include letters a and b. How on Earth am I to graph the given points to help me reach my goal here?

I would plot the four points above, and see what shape is the result.

2. I am confused about k. Above we see rho = k. Is k the density function?

$\rho$ is the density function (although, as per my other post, I prefer $\sigma$ for an area density). Since $\rho=k$, the density is constant.

 

[harpazo]

I would plot the four points above, and see what shape is the result.
$\rho$ is the density function (although, as per my other post, I prefer $\sigma$ for an area density). Since $\rho=k$, the density is constant.

How do I graph points that include variables in place of actual number?

 

[Ackbach]

How do I graph points that include variables in place of actual number?

Well, on your horizontal and vertical scales, instead of putting numbers, put the letters $a$ and $b$. I would assume they're both positive numbers, by the way.

 

[harpazo]

Well, on your horizontal and vertical scales, instead of putting numbers, put the letters $a$ and $b$. I would assume they're both positive numbers, by the way.

Can you set up the double integrals? I can take it from there.

 

[HallsofIvy]

You don't need to integrate anything. Your figure is a rectangle and the density is a constant. The mass is that constant times the area of the rectangle, kab. The center of mass is the center point of the rectangle, (a/2, b/2).

 

[harpazo]

You don't need to integrate anything. Your figure is a rectangle and the density is a constant. The mass is that constant times the area of the rectangle, kab. The center of mass is the center point of the rectangle, (a/2, b/2).

I want to practice double integrals. Can you set the double integrals for me?

 

[MarkFL]

To find the mass (using the calculus), we use:

\(\displaystyle m=kb\int_0^a\,dx=kab\)

For the center of mass, denoted by the coordinates $(\overline{x},\overline{y})$, we use:

\(\displaystyle \overline{x}=\frac{k}{m}\iint\limits_{R}x\,dA=\frac{1}{ab}\int_0^b\int_0^a x\,dx\,dy=\frac{a}{2b}\int_0^b\,dy=\frac{a}{2}\)

\(\displaystyle \overline{y}=\frac{k}{m}\iint\limits_{R}y\,dA=\frac{1}{ab}\int_0^by\int_0^a\,dx\,dy=\frac{1}{b}\int_0^b y\,dy=\frac{b}{2}\)

 

[HallsofIvy]

To find the mass (using the calculus), we use:

\(\displaystyle m=kb\int_0^a\,dx=kab\)

In excruciating detail, \(\displaystyle m= \int_0^a\int_0^b k dydx= k\int_0^a\int_0^b dy dx= k\int_0^a \left[y\right]_0^b dx= k\int_0^a b dx= kb\int_0^a dx= kb\left[x\right]_0^a= kab\).

 

[harpazo]

Thank you.