Use a triple integral in rectangular coordinates to find the mass of the region bounded by the planes x+y+z=1, x=0, y=0, and z=0 in octant 1, if the density at each point (x,y,z) is p(x,y,z)=kx
How do you set this up??
Always try to draw it first. If then when y=0, you have the line z=1-x in the z-x plane and if x=0, you have z=1-y in the z-x plane. Now just connect the lines in the x-y plane between y=1 and x=1. Try and draw those three lines in an x-y-z coordinate axis on paper then try to understand why the integral is (I believe):
It's been a while since I've done these, but I think it's:
The solid shape we are integrating over is a tetrahedron; having a geometric sense can help with determining the limits of the definite integrals. For general region R, the triple integral is
where we can integrate in any order. Here we will want to integrate with respect to x first, because p(x,y,z) only depends on x.
Anyway I get:
I'm pretty sure I got the upper limits right but not 100%, anyway you might find this site helpful, example 4 deals with a more generalized tetrahedral region defined similarly.
EDIT: maybe shawsend is right and it's easier to integrate dx last... it's been too long... sorry
EDIT 2: integrating with respect to x last is definitely easier.