Calculating Mass of Triangle Surface Given Density 4xz

[s3a]

Homework Statement

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.

Homework 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.

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.

Could someone please help me understand the step I am stuck on?

Any help would be greatly appreciated!
Thanks in advance!

 

[LCKurtz]

Homework Statement

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.

Homework 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.

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.

Could someone please help me understand the step I am stuck on?

Any help would be greatly appreciated!
Thanks in advance!

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.

 

[s3a]

Thanks!