Right. So let's take a look at the last example that I'm going to give you for this section. So um it's it's like you know there's no given sol I mean there's no given equation.
Okay. It says that find the volume of a pyramid with height h and square base of side b. So it's it's it's it's easy to sketch the solid because it's described has been described clearly.
Okay. So I have uh this solid right. So this is the solid.
Okay. Now the tough part is that there's no XY. So you cannot use um you know interal that's why we need to think of a way to to create an x ycoordinate.
So one of the most frequently used method is to find sort of a center. Okay. So I am going from the top let me see.
So the pin the top here. Okay. and then going down all the way going down and it will cross the base at the middle.
Okay. So we call that yaxis. We call that y-axis.
And then we go, you know, when we have the middle, the middle of the base, we call that the origin. Okay. So that's the origin.
Then we create the x-axxis. Okay. Then let me erase that back.
Okay. And then we construct more lines. Okay.
So once we have our y-coordinate and our x coordinate, we match the um the side going from the top to the bottom of the base. Okay, that one which is right here. Okay, which is right here.
And based on the given information, we know that the x intercept is 12 b and the y intercept is h. Okay. And we can do the same thing which is the other side.
And that's the one here. So that's another line. Okay.
And now you can we have the region. Okay. We have the region.
Okay. So this is the region. Okay.
So that's the region. Okay. But now of course we need to figure out the boundary of the region and we're going to do that later.
Now further analysis tells us that the crosssection of the volume. Okay. The cross-section of the pyramid horizontally in this case would be squared would be square.
Okay. And the um the yellow line here. So this yellow line, okay, is the same as this line.
And if we go back to the pyramid, it is the line in the middle. Okay, this line in the middle, but no matter what that is basically B. So that's the length B or similar.
Okay, similar. So make it simple. That's side.
Okay, that's the side. So our job is basically to figure out that in the language of XY. Okay.
So to do that um we call this tip A. Okay. We call this B and we call this C.
Okay. So I'll say as shown on the xycoordinate we have a coordinate is 0 h and b the coordinate is 12 b and zero. Okay.
Then the slope of AB equals okay. So that equals to H - 0 0 - 12 V. Okay.
So that give us oh negative 2h over b. Okay. -2h over b and the equation of the line ab is um we do have y intercept right we do have y intercept and y intercept is is h so y = -2h b x + H.
So that's the Y intercept and done. Okay. Therefore, the volume let's say this is big.
So the volume V will be the integral from 0 to H. area dy. Now why this is dy?
Well, it's right here. Okay, you can see that the cross-section goes like this way. So this is the cross-section.
So that's the cross-section and then we sum up this way. So going upward, right? We we sum up vertically then use dy, right?
we use dy and the region goes from y-coordinate from zero to y-coordinate h. Okay. So next what we really need right now is that um size square the area right the area of square is size square where the side equals to 2 * the x coordinate Okay.
So what I mean is that if this is x, right? So the x coordinate right here. Okay.
The xcoordinate of the point on the line where this x can be solved. Oopsie. Where this x can be solved by this.
Okay, so I'm going to solve it. y -2 h = -2h bx and further I have x = b 2 h y - h. So that's x and plug it in b 2 h y y - h.
Okay. And if I simplify it more, I'm going to have b over h y y - h and plug in which equals 0 hive b h y y - h I think and then square d y. Now my suggestion do not multiply this out.
It will it will be really tough if you multiply it out. Okay? Instead, you keep that and square this and keep the y - h² as one thing.
And we're going to have the anti-derivative directly as 1/3 y - h cub y. Okay? Because I'm going to use u as h I mean um y - h.
And then du is simply dy. [snorts] Then this guy will be uh well in terms of indefinite integral u² du. Okay.
And the anti-derivative is constant states 1/3 in cube. Okay. So that's the reason that's the reason.
And finally we plug in. So 1/3 b² h² time if I plug in h where's the negative? Oh that's okay.
So plug in h I'm going to have uh 0 cub minus if I plug in 0 I have this okay and simplify I have 1/3 b² h² * h cq and finally gets the answer/3 b ^² * h and that's the final answer and that's the area or sometime we can call that the formula for the volume not area so the formula for the volume of a pyramid whose base is a square okay so the key step okay the key step here is how to translate and interpret [snorts] the um solid in a language of region and cross-section. Okay. Of course with the help of um creating the um xy coordinate.
Okay.
