# Find the mass of the surface of the triangle with the given vertices and density.

1. Aug 2, 2012

### s3a

1. The problem statement, all variables and given/known data
Find the mass of the surface of the triangle with vertices (2,0,0), (0,2,0) and (0,0,1) if the density is 4xz. The solution is attached as TheSolution.jpg.

2. Relevant equations
Integration. Cross product. Plane equation: a(x-x_0) + b(y-y_0) + c(z-z_0) = 0 where <a,b,c> = n (vector that is normal to the plane) and where (x_0,y_0,z_0) is any arbitrarily chosen vertice.

3. The attempt at a solution
Having watched (1) and read (2),:

(1)
(2) http://www.math.oregonstate.edu/hom...usQuestStudyGuides/vcalc/surfint/surfint.html

I have successfully reached the dS = sqrt(3/2) dA step and for the next step, I'm supposed to multiply the density by the surface area in order to get the mass of the surface but, I just don't see how I'm supposed to get the (4x – 2x^2 – 2xy) part nor do I see how the variable density, 4xz, relates to that; I'm not sure that it should relate but I feel it should.

Any help would be greatly appreciated!

#### Attached Files:

• ###### TheSolution.jpg
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Views:
212
Last edited by a moderator: Sep 25, 2014
2. Aug 2, 2012

### LCKurtz

You have expressed the plane as z in terms of x and y. In the integral, the z in the density formula 4xz must be replaced by its value in terms of x and y on the plane.

Last edited by a moderator: Sep 25, 2014
3. Aug 2, 2012

Thanks!