Setting up Integrals to find mass and center of mass

[solomon684]

Homework Statement

Find the mass and center of mass of the lamina that occupies the region D and has the given density function ρ.

D is bounded by the parabola x=y² and the line y = x - 2; ρ(x, y)=3

Homework Equations

m=\int\int_D ρ(x, y) dA

The Attempt at a Solution

Basically I just need help setting up the integral for the mass, and I can get the rest.

What I did was set √x = x - 2, and solve for x, giving me x = 1 and 4. Therefore I made my integral for mass m=\int⁴₁\int^√x_x-2 3 dydx. However, I found online that it should be set up as follows:

Find where the curves intersect.
Since x = y^2, we have y = x - 2 = y^2 - 2.
==> y^2 - y - 2 = 0
==> (y - 2)(y + 1) = 0
==> y = -1 or 2.

For y in (-1, 2), note that x = y^2 is to the right of x = y + 2.

So, we can write the region as x = y + 2 to x = y^2 for y in [-1, 2].
(Integrate dx before dy.)

When you integrate each function you get different answers, so how do you know which way to set up the integral? I guess what I am asking is how do I know if the integral should be in the order dydx or dxdy, because both seem right here.

 

[Quinzio]

You should draw the graph of the 2 functions, then try to fill the inside region with horizontal or vertical lines. You want to find the simplest way, that is each line you draw to fill begins from the parabola and ends on the line.
Are both hor and vert equivalent ?
Now you should know how to integrate.

 

[solomon684]

Ok so in this example drawing horizontal lines gets you from the parabola to the line each time, where as vertical would go from parabola to parabola at some points, so because of that you integrate in the order dxdy right?

Now let's say it was bounded by to parabolas, or is in 3 dimensions. How do you know what order to do it in then?

 

[Quinzio]

Ok so in this example drawing horizontal lines gets you from the parabola to the line each time, where as vertical would go from parabola to parabola at some points, so because of that you integrate in the order dxdy right?

Now let's say it was bounded by to parabolas, or is in 3 dimensions. How do you know what order to do it in then?

bounded by to parabolas:
find a way to draw lines from parabola to parabola. Can't do it ? Split the region in two or more regions, then sum the integrals.

3d or higher: you apply the same reasoning. let's say we're in 3d. Volumes are bounded by planes or other 2d surfaces. Can lines be drawn from surface A to surface B ?
As an exercise you may want to find the tetrahedron volume bounded by:
x+2y+3z=10
2x+3y+4z=20
3x+4y+5z=5
2x-3y+4z=15

 

[solomon684]

Ok thank you, that was very helpful, makes a lot of sense now.

As for the exercise, I am kind of struggling with that. We have not done anything in class where it's bounded by 4 functions, I think the most we have done was 1 or maybe 2 functions that are in 3 dimensions and then the rest are planes in one or two dimensions.

My first thought is to add/subtract equations to come up with values of x, y, and z, but I'm not really sure if that helps

 

[Quinzio]

Ok thank you, that was very helpful, makes a lot of sense now.

As for the exercise, I am kind of struggling with that. We have not done anything in class where it's bounded by 4 functions, I think the most we have done was 1 or maybe 2 functions that are in 3 dimensions and then the rest are planes in one or two dimensions.

My first thought is to add/subtract equations to come up with values of x, y, and z, but I'm not really sure if that helps

Well, you don't need to do it in 5 minutes.
What you should think about is not the actual data (I put in some "random" numbers), but an algorithm to get to the solution, so that a computer could solve it.
If you know some kind of basic computer programming you could implement it. That will be a good passtime :)