Okay. So, let's take a look at a third example. We're doing the same thing.
Sketch. So, y = x cub. Oh, something like that.
Well, we don't we don't quite need a part uh underneath. Okay. And then y = to 1.
x = to 2. Not a good graph. Let me get a better one.
Maybe I need to exaggerate this a little bit. Oops. This should be blue.
Okay. And then I'm going to have a two here. Oopsie.
Let's get say this is two and then I get uh a one. Okay, so y equals what? Uh well, actually the graph looks like this.
Okay, so we don't we don't quite need the bottom one. Okay, so y = x cub and I have x = 2. And I don't think it would be hard to find this point.
So that's two and then 2 cub which is 8 of course and this is two one and this should be one one. Okay, it's very very easy to see it. Now I have the region.
So that's the region and I'm going to exaggerate the region a little bit. Okay, so this is the region. A not good, but um hopefully this is good enough.
Okay, now y = Oh my god. y = -3. So 1 2 3.
Oh my. Forget that. Let me get the region first.
So this is the region and then I have a y x. Um, it's it's pretty hard to draw it. Okay.
Yeah, that's the best I can do. Okay. So this is the rotational axis y =3 2 3 4.
Oh, this is tough. Yep, that is tough. Oh boy.
Well, I wish I have a ruler. Okay. So that's that's what I have.
Now the graph looks like this. Pretty ugly. Okay.
All right. So the cross-section looks like this. Okay.
Now make it simple. The cross-section is a ring. So this is the cross-section.
Okay. And I have Whoopsie. I have little R.
I have capital R. So I have a little R somewhere. Uh no.
Okay. So this this graph is just too hard. Okay.
So one, two, three, four. So I need to have it here. So it's even more uglier than what I thought.
This is what we have. Okay. There's the graph.
Okay. And the cross-section I will try my best. So the cross-section looks like uh here.
Oops. The crosssection looks like here. So in the middle.
Okay. So we have it like this. And this is little R and this is capital R and connection.
Okay. So we need a connection. Connection means this is little R.
Okay. And uh well let me make it bigger. Hopefully you can see it.
And then a capital R is like this. So this is capital R. Now those it's okay if it's not that clear.
However, it has to be very clear right here on the region. Okay. Now, this is little R.
Okay. And this Oh, no. I need the the rotational axis.
My bad. So this is the rotational axis. Okay.
So y = to -3. Now we're talking. This is my capital R.
Okay. And this is my little r. So my little r would be yt minus yb.
So top minus bottom and the top little r right is y = 1. The bottom is y = -3. So basically it's 1 -3.
Now that's the easy one. Okay. Now, here comes the capital R.
Still yt minus yb. Okay. But the yt is y = to x cub - y = -3.
Okay. Now, shall we use dy or dx? So this comes here and I don't think it is hard to see that we sum up the washer.
This is washer because there's a hole in this direction. So this direction. Therefore we have dx.
And since this is dx, we're going to have x cub minus - -3. So that's where my um little r and big r. And now we're ready.
Volume equals integral area dx. Okay, now let's get about because this is dx. Then we come here.
So we come to the region again. So this is my left bound and this is my right bound and that's the one. Okay.
So one and two because dx we're looking at the x axis. So x = to 1 and x = to 2 and I'm going to get the area where the area of the crosssection is pi r² - r² which is pi big r will get x cub + 3 I think squar - 4 squar so that's what we have and put it in I have x cub + 3 squar - 4 2 and the whole thing times pi dx and we u multiply things out and of course we can factor out the pi. So I have x cub becomes x to the 6 and then 6 x cub - 9 - I mean + 9 - 16 dx and solve it 1 2 x 6 + 6 x cub - 7 dx and then anti-derivative would be um 1 over 7.
Well, x 7 + 6 * 1 4 x^ 4 - 7 x evaluate from 1 to 2. Plug in I have 1 7 * 2 7. Oh, that's big.
+ 3 / 2 * 2 4 - 7 * 2 minus plug in one and the coefficient dominant and now no shortcut fortunately. So 2 to the 7 I don't quite know what it is. So 2 to the 6 I believe is 64.
So that is 128 over 7 + 24. Simplify that. That's uh 24 - 14 - 1 over 7 - 3 / 2 + 7.
Okay, so let's finish it up. 127 / 7 and then + 17. So this is 17 and then - 3 / 2.
This is a tough one. So up to you. You can use a calculator, you can use whatever.
So I I just too lazy to use calculator. So I have 2 54 over 14. Okay.
Uh do I need that? No. So let's do it this way first.
So this guy is uh 34 31 over two and then times the seven, right? That's what we have. 7 37 I hope this is right.
So 37 over 14. Hi. So that's what we have for that answer.
Okay. So we finished up this problem. And the reason I'm giving you this problem because next I'm going to give you exactly the same region.
Okay. So I'm going to use this. Okay.
But instead instead of this not uh rotation. Okay. Instead of this rotation axis.
Okay. So the rotation axis is y = -3. And this time I'm going to change it.
Okay. Instead of this I'm going to have uh oops here. Let me find it.
Is it down here? Yep, it's here. I'm going to dump it here.
Okay. So, this is number four. Instead of this y =3, I'm going to rotate it about x = to 2.
