If you have been working your way through these Summer Reading posts, then at this point, you know how to find “derivatives” for a nice collection of reasonable functions. Nothing too crazy or exotic — just the kinds of functions that you will meet early in AP Physics C. And you also know *why* we find derivatives: derivatives are slope-finding formulas.

For example:

**Velocity** is the rate of change of **position** with respect to time.

**Acceleration** is the rate of change of **velocity** with respect to time.

**Force** is the rate of change of **momentum** with respect to time.

**Current** is the rate of change of **electric ****charge** with respect to time.

For example, suppose you are given that an object moves so that it’s velocity as a function of time is:

**v(t) = t ^{2}**

You can easily find the acceleration as function of time by taking the derivative:

**a(t) = v’(t) = 2t**

So there are good, useful reasons for learning how to find derivatives. (In fact, let me remind you that to “optimize” a function, you usually begin by finding the derivative and then setting it equal to zero.)

**CAN WE PLAY THE GAME IN REVERSE?**

If I give you a function, can you find an “antiderivative”?

For example, let’s say: ** f(x)=x ^{2}**

Can we find another function that has f(x)=x^{2} as its derivative? If we know the basics, finding this particular antiderivative is not that hard:

**F(x)=(1/3)x ^{3}**

But you don’t have to take my word — go ahead an take the derivative! You will see for yourself that the derivative of F(x)=(1/3)x^{3} is indeed f(x)=x^{2}.

So F(x)=(1/3)x^{3} is an antiderivative of f(x)=x^{2}.

But it is not the only one! Try taking the derivative of any of these:

**F(x)=(1/3)x ^{3 }+ 5**

**F(x)=(1/3)x ^{3} – 99**

**F(x)=(1/3)x ^{3} + C** where C is a constant

So if you can find one antiderivative, you can find a whole family of them. Still, most times, you only need to find one.

[Caution: your math teacher will likely take off a point or two if you leave out the “…+ c” when finding an antiderivative.]

**IS IT ALWAYS THIS EASY?**

Well, no. Some of them are easy. But others…not so much. You will spend a substantial amount of time in math class building a collection methods for finding antiderivatives for an increasing variety of functions. And some of these methods are quite subtle. For now, I want to make two points:

**In our AP Physics class, when the antiderivatives are hard to find, you can have a little help: your TI89s or N-Spires know how to do this. So does Wolfram Alpha.**

**Nobody would be willing to invest so much effort in finding antiderivatives unless there was something really useful about them! **

**A NEW NOTATION**

If we are going to be using antiderivatives, we should have symbols for writing them. To indicate that F(x) is the antiderivative of f(x), we write:

This is just another way of saying that F'(x) = f(x). And, in addition to calling F(x) an antiderivative, we also sometimes call it the “indefinite integral of f(x)”.

**BUT WAIT…**

On the other hand, there is another symbol that looks almost the same but means something *completely* different:

This symbol is called the “definite integral” (as opposed to “indefinite”). It is not a function at all! Just adding values at the top and bottom of that strange elongated s-shape gives the symbol a whole new meaning:

The definite integral is the area of the region bounded by the function, the x-axis and the two “limits of integration”: the vertical lines defined by x=a and x=b.

For example, here is a definite integral:

And here is the area that definite integral is referring to:

So now we have two very different ideas that have been given very similar symbols and names. It would be easy to get these two things confused…

There is the * INDEFINITE INTEGRAL*:

This is a FUNCTION which has f(x) as its derivative.

And then there is the * DEFINITE INTEGRAL*:

This is an AREA — a numeric value.

And now the big question:

**Why do mathematicians use these similar names and symbols for these two very different ideas? Are they secretly related in some FUNDAMENTAL* way? **

That is the topic of the next post.