Triple Integral: Evaluating Limits

[mvpshaq32]

Homework Statement

Evaluate the triple integral.
∫∫∫xyz dV, where T is the sold tetrahedron with vertices (0,0,0), (1,0,0), (1,1,0), and (1,0,1)

The Attempt at a Solution

I'm having trouble finding the bounds. So far I'm integrating it in order of dzdydx with my x bounds as 0-1, my y bounds as 0-x, but I'm not sure how to find the z bounds.

 

[lanedance]

if you integrate over z first, it will be bounded by function of both z & y, representing the top plane of the tetrathedron

 

[lanedance]

Homework Statement

Evaluate the triple integral.
∫∫∫xyz dV, where T is the sold tetrahedron with vertices (0,0,0), (1,0,0), (1,1,0), and (1,0,1)

The Attempt at a Solution

I'm having trouble finding the bounds. So far I'm integrating it in order of dzdydx with my x bounds as 0-1, my y bounds as 0-x, but I'm not sure how to find the z bounds.

similarly your y bounds are not y=0 to y=x, but will be y=0 to y=1-x, to see this look at the line formed by the top surface of the tetrahedron in the xy plane

 

[mvpshaq32]

similarly your y bounds are not y=0 to y=x, but will be y=0 to y=1-x, to see this look at the line formed by the top surface of the tetrahedron in the xy plane

Sorry, I'm still confused. How would I integrate over z first?
And I am not seeing the line y=1-x formed. I plotted (0,0,0), (1,0,0), (1,0,1) and (1,1,0) onto a xy graph so the points would be (0,0), (1,0), and (1,1) giving me the line y=x or is that not how I approach it?

 

[lanedance]

apologies you are correct, so the integral should be
$$int^1_0 (int^0_x ( int_0^{f(x,y)} dz) dy ) dx$$

now you just need to find the upper bound for the first integral f(x,y) which is the upper bounding plane of the tetrahedron.