Volume

Let me try my best to draw this neatly.

So let's think about it.

So if I had a little cube here in the volume under consideration, that's a little cube you consider that dv.

And that little cube has side length 2.

Well, they also tell us that the volume is 405.

And that's that the ratio between the volumes let me write this down that the ratio between the volume at state B and the volume at state A so the ratio of that volume to that volume is equal to, in our Carnot cycle, is equal to the ratio between the volume at state C.

So how do we figure out the volume under a surface like this?

So dy, which is this. dy times dx, dx times dz.

In units, what are the dimensions of the box?

Now if I multiply this whole thing times dy,

That makes sense because surface area is a 2 dimensional measurement. We think about how many sq cm can we fit on the surface of the cylinder.

And that seems pretty reasonable for our reality.

Let me draw it.

These are the same rectangles.

This imbalance, if we want to stabilize climate, means that we must reduce CO2 from 391 ppm, parts per million, back to 350 ppm.

You either multiply the length of the rectangle times the entire width.

But you put one there. Why?

So what's the volume of this shell?

It's kind of a constant y.

And we go straight through the centimeter that distance right over there is 14 centimeters.

And then it's going to be times the spanning vector itself.

But that's the intuition of what we're going to do.

So this is the surface defined by this function.

And that width is dx, so it's going to equal this times dx.

But I'm going to make it a little bit more complicated.

Whereas the other three piles were dead, dark and stinky, and the PAH's the aromatic hydrocarbons went from 10,000 parts per million to less than 200 in eight weeks.

So if we have this unit step function, this thing is going to zero out this entire integral before we get to c.

We piled the carcasses and burned them.

Then he cuts out a little cube of side length 2 from one of the corners of the cube.

Well, we know that the surface area of each cube is going to be 6x squared, where x is the dimensions of that cube.

A cube with side x has length x, width x, and height x.

So for this cube, the surface area is going to be 6 times 2 squared.

So what would be the volume of that sliver?

With that out of the way, let's just apply this radius being 7 centimeters to this formular right over here.

And that's taking the integral.

length times x times 9.

That's, literally, the volume of the box.

So let me does this, it's going to be 8 cm times 8 pi cm. And thats equal to 64 pi. 8 times 8 is 64.

That half of the rhombus

So this thing is equal to 1.

Z is equal to 2.

Right?

So this expression, as it is written right now, is the volume above this rectangle.

But what would that give us?