All right. So now let's take a look at some examples. So the first example is like this.
Find the volume of the solid whose base is bounded by the graph of y = x + 1 and y = x^2 - 1. The cross-section perpendicular to the xaxis is a square. Okay.
Is a square. Now, um I found the the solid, the gram of solid, but generally speaking, you don't quite have to really sketch this because it's it's really hard. So, what you really need to know is the region.
So, first sketch the region. Uh let me see. Okay.
So I have this y = x² - 1 and then I also have uh let me use blue the line y = x + y. Okay, I think this is the y. Okay.
So, y equals to uh I'll write an equation later right here. So, this is y = x + 1 and this is y = x^2 - 1. Okay.
Now the third graph would be the cross-section. Now the cross-section is easy square right. So this is crosssection.
Okay. And this is the region as the base. So this is the region as the base and this of course is the solid and generally speaking it's hard you don't have to do that but for the region and the cross-section you have to have it.
Okay. Now of course uh the cross-section is square. Therefore the area is side square.
Right now the most important step is where is this side? Okay where is this side? Now it says that the cross-section perpendicular to the x-axis.
So perpendicular to the x-axis meaning this. Oopsie. Meaning it looks like this.
Okay. So those are the the base. Okay.
Those are the base. Okay. It's important to understand this because here when we sum up the cross-section the direction is [snorts] this direction, right?
We have um one, two, three, four. So, a lot of them, right? So, we sum them up this way.
Therefore, we have dx. Okay. All right.
Now take a look at the cross-section again or basically the side. Okay. So this is the side will be top minus bottom.
So what I mean is that you know this is the top this is the bottom. Therefore, we are going to have yt minus yb where yt is the line y = x + 1. Right?
So, y = x + 1. And for yb is y = x^2 - 1. And pretty much we're done.
Right? So this tells us it's x + 1 - x^2 - 1. Okay.
And then once we have the side. Okay. So once we have the side, right, we have the side, we plug it in and that's the area.
Okay. Therefore, we can solve it now. Okay, so the volume of the solid V equals area of the cross-section DX because of here right now since we are doing dx then we need to figure out the region like the bounds for x and the bounds come up here okay and we can easily see that those are the points and how to solve it well where the bounds determine by this system - 1 + 1 y = x^2 - 1 and we solve it.
So x^2 - 1 = to x + 1. And let's continue solving it. - x - 2 and I have x - 2 x + 1 and further x = 2 or x = to -1.
These two values would be the bounds which equals Then v = -1 to 2. That's the area would be x + 1 - x² - 1. So this is the side square the x.
There we go. Uh let's do it here. I'll do it side square dx where side equals the y top - y bottom and the top is the line x + 1 and the bottom is x² - 1 that's the parabola and I'm going to simplify this.
Okay. And unfortunately, there's no shortcut. Um, that's pretty heavy.
The calculation is pretty heavy. So, multiply this thing out. This is hard.
Oops. I have x^ 4 + x^2 + 4 - 2 + this plus this and then plus this and further simplify at uh I have x to the 4 x cub so I have x cub here - 2 x cub and let me see x² I have x square here I have x² here so I have uh 3 x² and then x term I think this is the x. [snorts] Okay.
So, plus 4x and then constant. Yeah. And then constant term.
+ 4 from -1 to 2. Anti-derivative 1 over 5 x 5th - 2 1 / 4 x 4 - 3 1 3 x cub + 4 1 / 2 x² + 4x. evaluate from -1 to 2.
And to that no shortcut 1 over 5. Now this time myself for communion I'm going to have this -1 to 5th minus 12 to 4 - -1 to 4th and then minus 2 cub -1 cub and then 2 2 -1 2 and then 4 2 -1. So uh well I'm going to explain this a little bit later.
It's just the way that I am um simplifying my calculation. So I have 32 + 1 over 5 and then minus to the 2 to the 4th is 16. So that's 16 - 1 / 2.
And then I have 8 + 1, that's 9. Uh yeah, a - one. And then I have 4 - one.
And then I have uh 4 * 3. And too bad there's no shortcut once again. So I'm going to have 5 33 minus 15 2.
Now what we have here? So that's 9. That's three.
I have -9. I have six. I have 12.
So I have + 9. Tough. And solish.
I have a 10. 66 - 75 + 90 that is um 15. So it's 81.
So something like that. Okay. Now well I just explained this a little bit.
It's just the way that I I simplify it. So what I'm doing is this. Okay.
So 2 to the 5th - 2 uh - 12 2 4 - 3 I mean 2/3 + 2 2^ 2 - 4 * 2 - 1 5 - 1 5 - 121 one to the fourth and then da da. So what I did actually what I'm doing here is this part and this part. Okay, I put them together because they are going to share the same denominator.
So that's just one way to um well to me it's simplifying my calculation but you don't have to. Okay so that actually give us the answer pretty tough. Okay.
So next I'm going to give you one example with triangle and then one example uh one example that require us to set up the coordinate system all by ourself.
