Thread: Help setting up triple integrals from 4 points

The topic pretty much says it all. I have to evaluate the triple integral of xyz for a tetrahedron with vertices (0, 2, 0), (1, 0, 0), (4, 1, 0) and (1, 0, 1) then I have to do it again with vertices (0, 2, 0), (1, 0, 0), (4, 1, 0) and (1, 1, 1). I shouldn't have a problem solving them once I get them set up, my problem is that my professor did not explain how to set up a triple integral from coordinates. Can someone please give me step by step tutorial on how to go about setting up these integrals?

Sure. What is your narrative for a single definite integral of y = f(x) with respect to x? Mine is...

I walk along the x-axis from the lower limit towards the upper, and at each point I turn left and note down the distance away (along the x-y plane in the y direction) of the curve y = f(x). The integral, e.g, for the 'curve' (though straight line)
...



... is the total of all these values. Hopefully you can intuit this total as the area under the curve. (Do imagine visually, e.g. sketch, all of the curves I refer to.)

Or we could get the same result from observing (at each point on the x-axis journey looking in the y direction) the difference between the distances to each of the two curves and ...



But let's turn either of these into a double integral of z = 1 (with respect to x and y) ...

I walk along the axis from the lower limit towards the upper, and at each point I turn left and this time go on a detour in the x-y plane in the y direction, from the x-axis to as far as the curve for f(x). At each point along the detour I note down a number 1. Then the integral of all those ones...



... or else, if each detour in the y direction goes from the curve for up (in the y direction) to the curve for , then this integral of all the ones...



... will be the same total as before.

Now, every time we noted down 1, i.e. at each point on our detour along the x-y plane in the y direction, we might have been looking up above us in the z direction (the x-y plane being horizontal ground) and observing the distance above us of the plane z = 1.

What if, above us at each of these points, we saw not the plane z = 1 but the plane ...



And here you can probably see that...



... would be want to produce a different volume.

Again, we could turn this into a triple integral by going on a third detour, in the z direction, and observing, at each point on the detour, a distance of 1 in some 'fourth' dimension (that's difficult to visualise as a new spatial dimension, but might be density or temperature etc.)...



And what you have in the first section (west of the plane x = 1) of your first tetrahedron is the same volume, but the density to note down at each point is xyz instead of 1...



That's the basic approach.