Instead of blowing the balloon up like a party balloon, they use a giant flame to heat the air inside that envelope.

Air is a mixture of gases.

The air is a medium for sound.

But they are really acting mostly as conduits.

Hopefully, that gives you a little intuition, and so you'll never forget that pressure times volume is constant.

Molar Gas Constant

Air is a mixture of gases that we cannot see.

Air is a mixture of several gases.

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

Let me try my best to draw this neatly.

Here you can see that knife went through the heart.

Air is a mixture of various gases.

And, this notation right over here, this whole expression, is called the indefinite integral.

So let's think about it.

Volume

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

What are you gonna do on your own?

I mean, if I just go through some of them now

If that's the case, then what is v?

And we have its derivative sitting right there.

We now turn to Measurement of Cosmological Parameters. The first one of which, is

Please fill in the parameters for the gas particles.

And that little cube has side length 2.

Air.

Now let's go the other way around. Let's think about the antiderivative.

Anti derivative.

I say, if I have an expression, and I have its derivative multiplied there, then I can just take the antiderivative of the whole expression.

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.

You might think that that's not a surprise.

Here you can see that knife went through the heart.

If f prime of t is equal to cosine of a t, what is a potential f of t?

I'll keep the improper integral with us the whole time.

You wanna tell me what the hell's going on here? What are you on?

Only change, four times the area.

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

The air of hills is cooler than that of plains.

little bit, my y looks like a g. y minus y to the third over 3 is equal to x to the third over 3 plus the constant when we took the antiderivative of the x, minus the constant of the antiderivative when we took the y.

Now, if we take the integral of both sides of this equation, we get uv is equal to the antiderivative of u prime v plus the antiderivative of uv prime.

The air of Sydney is cleaner than that of Tokyo.

Because remember, we spend 90 percent of our lives indoors.

Let me draw it.

These are the same rectangles.

We're just taking the derivative backwards.

Now if I multiply this whole thing times dy,

And then, well, you would end up saying that the antiderivative is just the natural log of u.