Now let's look at more examples where the rotational axis is neither x-axis nor y ais. So look at this example where the region is bounded by y = 4x - x² and then y = 0 about x = to 1. Um I just realized that my graph is wrong.
Oh anyway. So I'll fix it. So um I have the region say this is the parabola y = to 4x - x².
So the way to sketch it is that I have x intercept x = 0 and x = 4 and then the vertex would be two and four. Okay, I got this point, this point and this point. So connect them as a parabola up and down and then y = 3 is the line horizontal line.
So I realize that my second graph this graph is not what I want because the rotational axis is x= to one. So I was I was thinking this. Okay, I was thinking this which is force is not right.
Right. Okay. So, let me erase this and then I get a graph again.
Right. So, I have this graph. One, two, three, four.
One, two, three, four. I have a point here. I have a point here.
And I have a point here. So, this is the graph. And I'm going to highlight it.
And then the rotational axis uh let me use yellow. So this is the rotational axis. So x equals to one.
That means I'm going to copy this. So I'm going to copy this and then get another one. All right.
right here. And then a cylindrical shell. Maybe I need a bigger one.
Okay. Cylindrical shell looks like this. So this is the cylindrical shell.
Okay. I mean I can I can I can have something like this, right? So I can have something like this or I can have something like this a little bit longer, you know, higher.
Okay, so I can have those shown. But anyway, what's next is you know identify the R and this is my R and this is my H. Okay.
So keep that in mind and thickness. So the thickness is this. So this is the thickness which give us delta x.
Okay. So once we analyze the graph and again I am requiring both graph. So after we analyze the graph we know that so by the method of cylindrical show we have volume equals 2 pi rh and found we need to solve it.
Okay. Where CD can be solved by this. So solve this intersection.
- 4x + 3 = 0. Actually, I know it's one and three, but I just want to um show you and actually I will require you guys to really solve it. Oops.
One and three. Okay, this is how we can solve it. So we have one and three and then I have 2 pi.
Now r. So here comes the r. So r would be let me get the rotational axis.
Ah the graph is not that good. I'm going to erase it. Okay.
So, the graph looks like this. Okay. Now, I maybe it would be a good idea for me to make it bigger.
Okay. to make it bigger or well let me do it this way. It's a little bit bigger.
Okay, so this is y = to 4x - x² and this is x = to 1 and r is like this. So this is r okay where r equals to the curve minus x = to 1. So I'll do it this way.
And a curve would be x curve the x value of the curve minus the x value of this line. Of course it will be one later. And is it a Yeah.
Okay. Then I'll just say x - one. Okay.
X - one. Okay. And the height is right here.
So this is the height. Okay. So height would be our Y curve.
- y = 3 and the curve is 4x - x^2 in in in terms of x and then y = 3 that's for the um the bottom. Okay. So you can think about this as top minus bottom.
You can think about this as right minus left. Okay. So now we can get the radius as x -1 and the height would be 4x - x^2 - 3 and then dx.
Oops. Okay. And then continue with the calculation.
I have 2 pi 1 3. I have -x cub + 4x^2 - 3x and then plus x²ar - 4x and then plus three and then dx. Uh sorry I didn't quite organize this well.
All right. And then continue with the calculation. I have -x cub + 5x^2 - 7x + 3 dx.
I mean please verify my calculation. I I did this first and then this. Okay.
So one two. Okay. and continue with the anti-derivative.
I have 1 /4. Oh, the calculation here is not fun. X cub - 7 / 2 x² + 3x evaluate from 1 to 3.
So, this is tough. And I'm going to have 81. So, I'm going to do it directly.
And then I have uh 45 and then minus 63 over two + 9 and then minus plug in one the coefficient dominate and then get the final answer. Uh not easy. Um so I'm going to get the answer directly somewhere here of -81 over 4 - 126 over 4 + 54 and that give plus -81 - 126 + so times 4 there 26 okay and whatever I have it's uh 81 right so 207 which is 9 so 9 over 4 oh this is nice minus and the other one would be 12 right so 12 FCD is 12 -3 + 12 20 - 42 + 36 I have 50 6 - 45 I get nine in.
Okay. And then do the calculation with three 18, right? So 18 over 12.
Oh. Oh, this is nice. No, this is even better.
Uh 3 over2 I think. This is nice. 3 pi.
Okay. So heavy calculation but the good thing is that you know um at least for the exams I probably ask you to do the setup only. That's the good news.
