Calculus 3 project – any and all help is appreciated.(adsbygoogle = window.adsbygoogle || []).push({});

We've just gone over center of mass with double integrals, so it's a bit peculiar to see this project feature only one integral. I went over that in calc 2 – and as a result, know how to calculate it that way.

However (as you will see in the formula below), I don't think he wants me to calculate it that way. I have no idea how to proceed and the book is of absolutely no help.

Anything you could do to help me out would be fantastic.

And here's the problem:

1. The problem statement, all variables and given/known data

There are three type of laminae we are trying to find:

1) A triangle with sides 3", 4", and 5" respectively.

2) A semicircle

3) A "horseshoe", i.e. a half-ring, whose inner and outer edges are composed of circular arcs of radii 3" and 5" respectively.

Definition 1:

The first moment of a planar mass distribution about a line l is the integral over the region of the (area) density σ(P) times the distance of P from the line. For example, the first moment about the line x=k is

[itex]M_k=∫_R\left(x-k\right) σ\left(P\right) dA[/itex]

Definition 2:

In order for a line to be a balancing line, the first moment about this line must be zero.

2. Relevant equations

See the M_k above

I found, I believe, the equations for the laminae:

For the triangle,

[itex]y=4-\frac{4}{3}x[/itex]

For the semicircle,

[itex]x^2+y^2=1[/itex]

And the horseshoe,

[itex]x^2+y^2=25[/itex]

[itex]x^2+y^2=9[/itex]

3. The attempt at a solution

Welp, I'm lost. As I said, I can figure it out easily using the calculus 2 method of [itex]x_{cm}=\frac{M_y}{M}[/itex], but I'm not sure the way he wants it done.

What's the density formula? Is it just the equations I listed above? What's this (x-k) nonsense?

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# Homework Help: Finding the center of mass of laminae

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