00:01
Okay, so we're graphing, i mean, we're finding the integral of this function of three variables over a volume or a solid.
00:14
So we don't care what this z minus x looks like.
00:19
In fact, we can't draw it because it's in four dimensions.
00:23
Okay, and we're graphing, we're integrating from the x, y, plane, up to this z.
00:29
So the bottom to top so we don't really care what this looks like either this whoops this all we care about is what's in the xy plane because the rest of it has already told us we're gonna integrate zero to x plus y squared z minus x d z all right so let's look in the x y plane and we have vertices zero zero because we don't care about that.
01:05
Those are all zero.
01:06
That's just putting us in the xy plane.
01:09
1 -1 and 0 -1.
01:17
All right, so we're integrating over this triangle, and this line is y -quels -0.
01:24
This line is x -equals 1, and this line is y -quels -x.
01:32
All right, so now we can either do d -y -first or d -x first.
01:37
It doesn't matter.
01:38
If you do d -y -y -you're going to go from y -equal -0.
01:41
0 to y equals x and then you're going to stack those up from this x to this x so 0 to 1 d x okay or the other way here to x plus y squared z minus x d z we're going to do d x first then we're going to do from y x equals y to x equals one and then we're going to stack those up from the bottom to the top zero to one okay so either way so let's just do the the one i did first so we're going to integrate with respect to z and we get z squared over two minus x z from zero to x plus y squared z y d x all right, i'm going to do all this algebra over here on the side so they don't have to keep writing all the integrals and junk.
03:00
Plus, i can mess with it a little bit.
03:05
I'm going to factor out a z.
03:07
So i have z over 2 minus x...