Well, let's continue with uh more examples. And this time the example is about a solid while the region generate generated by rotation about xaxis. So we have two curves x + y = 4 a straight line.
x = y^2 - 4 y + 4. That's a parabola open to the right. So let's look at the graph.
I mean if you cannot redraw them at once, you can draw it one by one, right? So the straight line and the parabola. Now once I have the parabola, I add the straight line and you can see the region.
Now once I isolated the region out, we're going to rotate this about x-axis and we get this. Okay. Now the cylindrical shell would be sideway because of the rotational axis as x-axis.
Now I'm going to copy this line to the region which is right here. Right? So this would be our h.
Right? So this is h and this will be r. So this would be r.
And a cylinder looks like this. Okay, we can analyze that r is simply y and h you can think about this as line minus uh parabola, right? You can think about this.
Okay. And then uh what else? Oh, thickness.
So the thickness mean I can exaggerate this a little bit. So the thickness is like well it's like this. Okay.
So this is the thickness. Therefore you can see that is delta y. Okay.
So dy another way another another way to say it is simply delta y or sometime we can think about this as dy. Therefore, our integral is in terms of y. So, let's set it up.
Uh, I'll just copy the form here. So by the method okay so I'll just say that you know um by the method of um cylindrical cylindrical shell we have a volume equals to integral 2 pi rh and then dy. This time it's dy.
And then we can set up the bounds. So I'll call that c and d where cd can be calculated by this. So we solve this system of equation.
I mean you you have to show this work otherwise I don't know you know I don't know where you get the answer from. So basically I substitute this guy here. Okay.
And then simplify and solve it. I have y^ 2 - 3 y = 0. So y y - 3 = 0.
Therefore I have y = 0 and y = 3. So this means v equals the integral from 0 to 3 2 pi. So r is my y and h would be the line which is uh uh 4 - yus y^ 2 - 4 y + 4.
Now here I want to explain it a little bit more. Okay, here. So h equals to y the y well no it is the x value of the line minus the x value of the parabola.
Okay cuz it it's like we we were looking for a line horizont I mean the length of a line horizontally then we're using the xcoordinate uh we're using the x coordinate and do the subtraction. Okay. So the line which is um x + y = 4.
Well, this of course is not the right thing to write it, but I'm just saying we're using this and then we're solving it. So x = 4 - y and x = y^ 2 - 4 y + 4. So this is just a scratch work.
Okay. So this is how we analyze it and that's the reason I have this guy and this guy and then simplifying it oops I have um 2 y sorry 2 pi so take it out 0 to 3 I got y 4 - y - y^ 2 + 4 y - 4 d y. Simplify further and we have - y3 + 3 + 3 y^ 2 d y.
Okay. So this give us 3 y and anti-derivative -1 over 4 y 4 + y cub and then evaluate at 0 and 3. When we have zero, everything goes to zero and then I have 3 to the 4 + 3 cub uh which is well of course minus zero right.
So the way to do that I will factor. So what I'm saying is that uh I mean you can use calculator that's for sure. So this would be uh 3 to the 4th pi over 2.
Okay. The calculation goes like this. So 2 pi 3 cub and then 4 over 4 - 1 over 4.
And this gives you 3 over 4. So I have 3 to the 4th. Now 3 to the 4 is 81.
Okay.
