A Magical Theorem, Part I: Antiderivatives and Areas

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) = t2

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.)



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

For example, let’s say:  f(x)=x2

Can we find another function that has f(x)=x2 as its derivative? If we know the basics, finding this particular antiderivative is not that hard:


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)x3 is indeed f(x)=x2.

So  F(x)=(1/3)x3 is an antiderivative of  f(x)=x2.

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

F(x)=(1/3)x3 + 5

F(x)=(1/3)x3 – 99

F(x)=(1/3)x3 + 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.]


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! 


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)”.


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…



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.


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