4 3 PI times the radius cubed... 1.26...

So what is this equal to?

Let me write a nicer integral sign.

I'm focusing so much not to make careless mistakes.

These adjectives are all positive.

This is essentially just this is the indefinite integral of x squared plus 1.

So what's the integral of this thing?

A million years ago?

If I were to tell you that I have a two dimensional space that has a finite area, so it's not infinite and also has no edge this once again, when you first look at it, it seems difficult how do I construct something that has a finite area but still has no edge?

We have a similar base, so this is equal to what?

Pop up a window when a disk gets critically full

That is a nice question, and it's great to see students enthusiastic about extending the course project.

So we've added in support for 2 new optimizations.

If we take out the constants from inside of the integral, we get e to the minus sc times f of c times the integral from 0 to infinity of f of t minus c dt.

And we'll do more on that intuition later on.

Fair enough.

Now, chemically speaking, this is equivalent to lowering the volume of a reaction vessel or a reaction mixture.

Your Dirac delta function is a function of t, and because of this, if you ask what's the limit as tau approaches zero of the integral from minus infinity to infinity of d sub tau of t dt, well, this should still be 1, right?

So this is going to be equal to c1 times the integral from 0 to infinity of e to the minus st, times f of t, d of t, plus c2 times the integral from 0 to infinity of e to the minus st, g of t, dt.

It's almost impossible to do that with your hand except in shorthand, and then communication is limited.

Taking the integral of this thing from minus infinity to infinity, since this thing is this thing, it should be the same thing as taking the integral of this thing from minus infinity to infinity.

The Laplace transform of f of t is equal to the integral from 0 to infinity of e to the minus st times f of t dt.

We're going to have to take this whole thing and evaluate let's just say that y is the antiderivative and then evaluate it from infinity to 0.

To calculate the capacity, multiply the length by the width by the depth.

As the temperature increases, gases move away from one another, and the volume increases as well.

So the Laplace transform of some function of t is equal to the improper integral from 0 to infinity of e to the minus st times our function.

It peaks in early adolescence.

It's one sided. It's a bottle whose inside is its outside.

Okay, so what are some possible solutions to users entering bad data in our form?

And I want to make something really clear

Laplace Transform.

So that's minus 1 over s e to the minus st, times v, times cosine at, minus the integral from 0 to infinity.

What it is now is mud, and the area of the ocean floor that has been transformed from forest to level mud, to parking lot, is equivalent to the entire area of all the forests that have ever been cut down on all of the earth in the history of humanity.

They were a highly intelligent species.

And you hear the words scarce resources a lot when people talk about economics.

And, just for the record, one mad scientist is my limit.

I'll just use that old integral sign.

Our cities, I would put to you, are stockpiles of these surplus capacities.

So how do I do that?

So we really have a bound on the utility here.

I've been accused of murder, and you're going to exonerate me.

Great bulk does not always mean great weight.

And we can just use the definition of the Laplace transform, so this is equal to the area from 0 to infinity, or we could call it the integral from 0 to infinity of e to the minus that's just part of the Laplace transform definition times this thing and I'll just write it in this order times f of t times our Dirac delta function.

And let's get this y over onto the dy side, because it will be easy to take the antiderivative then.

So we have to estimate that some way.