In a question about finding the mass of a Tetrahedron
The question given is this: Find the mass of a tetrahedron with corners at (0,0,0),(π/4, π/4, 0), (0, π/4, π/4), and (0, π/4, 0) whose density at point (x,y,z) is sin(x).
Recall that the mass of a region is the integral of the density function over the region.
Give your answer as an exact expression.
I really just guessed the bounds based off a similar question. How do I actually go about finding these bounds? I can't visualize this object very well and I am not to sure how to draw something like (π/4, π/4, 0), (0, π/4, π/4), (0, π/4, 0).
Could someone maybe describe the process of finding the bounds? The actual integration steps are straightforward.
$$ \mathbb{P}: x+y+z = \frac{\pi}{4} \rightarrow z = \frac{\pi}{4} - x - y $$ $$ \mbox{XY}: y = -x + \frac{\pi}{4} $$ $$ m=\int_0^{\frac{\pi}{4}}\int_0^{\frac{\pi}{4}-x}\int_0^{\frac{\pi}{4}-x-y}\rho(x,y,z)dzdydx $$