Find mass of triangle A(1,1,2), B(3,1,4), C(2,2,5). Density(x,y,z) = y.
My attempt:
Equation of plane A(1,1,2), B(3,1,4), C(2,2,5) is $z=x+2y-1$. $$ M = \int_1^2 dx \int_x^1 dy \int_{x+2y-1}^{x+2y-1} y dz + \int_2^3 dx \int_{-x+4}^1 dy \int_{x+2y-1}^{x+2y-1} y dz $$
Your triangle in 3d space projects onto a triangle in the xy plane as shown in your diagram above.
How does the area of the triangle in 3 dimensions compare to this projection?
It is $\sqrt 6$ times bigger. (Find the area of each and compare.)
Or find the secant of the intersection of the lines normal to both the plane containing the triangle, and the xy plane.
$\iint (\sqrt 6) y\ dy\ dx + \iint (\sqrt 6) y\ dy\ dx$ with the limits of $x,y$ that you have worked out above.