They asked us to integrate this function or find the volume of the solid underneath this function over this region. So the function is four minus X minus why? And the region goes from um how did they give us the region? I guess they gave us two Ears and a bunch of lines. So we have uh this line is where we hear hawaii cause four minus X. And this is X equals zero. And this because why it goes zero. That's wait a minute. Uh Oh yeah yeah so that's you know basically this I wrote it in set notation. So why goes from 0 to 4 X. Or form hopes that it will be four minus X. Wild Coast. Um 0 to 4 minus X. And X. Co X is greater than zero. Um We don't really have to we could put X's great between zero and 4 if we wanted to. Um But again that pinches off here so um yeah no let's see here actually I think probably ought to write that explicitly out because theoretically um you can have kind of this region out here, right? Which would kind of be, you know, if X went if you had X between zero and five, I guess you would extend this out and you have this kind of double two triangles anyway. They said they said we don't have to plot with this um volume looks like, but I did that anyway because I don't know, I wanted to see what it looked like. I knew this was a plane and we have a plane and then over find over a triangle. So we have this kind of wedge here. Okay. Doing the integral. Hopefully we can, you know, I'm not going to go through all the details here because this is a pretty simple integral. And hopefully you can do this pretty easily by now and you get 35 or 32/5, which is interesting because if you say, okay, well what is the volume of this cube? And that's four by four by four, which is 64. So if you take 1/10 of that, look at this, which is kind of surprising, I think This is only the 10th the volume of this volumes. People have very poor intuitions about volume. Um You know say if you look at the sphere that has a volume of one cubic meter and then a square that has the same volume, they don't look the same at all. Basically people have very poor instincts and intuition on volume. So the fact that this is 1/10 of the volume of this of the total cube, it seems a little surprising. But again, I can see it, you know, I could see how that could possibly be.