Question

Find the mass M=∭ ρ(x,y,z)dV of the solid in the first quadrant (namely x≧0, y≧0, z≧0) bounded by the coordinate planes and the graph of x+y+z=1 if the density is given by ρ=x+2y.

Answer 1

The integral would be
\[\int\int\int_D\rho(x,y,z)~dV=\int_0^1\int_0^1\int_0^{1-x-y}(x+2y)~dz~dy~dx\]

Answer 2

Thank you, I wasn't sure of the boundaries.

Answer 3

Given that \(x,y,z\ge0\), you can reason that for \(x+y+z\) to be equal to 1, the greatest possible values of the variables must be 1, and the lowest 0.

Start off by using the lowest possible values for any two variables, then solve for the remaining. For example, \(x+0+0=1~\Rightarrow~x=1\).

Another way would be to draw the plane, but that could be difficult, generally. This particular plane isn't too difficult to picture mentally, though.

Answer 4

And you're welcome!

Answer 5

Middle limit needs work. It's a tetrahedron. Should be 1-x on the top.

Answer 6

@tkhunny, ah thanks for catching the error!

Answer 7

@Dark_Fantom, the difference between my limits and tkhunny's correction is as follows: