Langara College - Department of Mathematics and Statistics

## Antiderivatives

Antidifferentiation is the process of finding functions with a given derivative. Eg finding distance travelled from a known velocity history, volume from flow rate, or net worth from records of income and expenditure.

A site in Austria allows you to "draw" a graph by hand (though it may take a bit of practice to get exactly the shape you want) and see an antiderivative displayed at the same time. (You can easily check by eye that the slope of their blue graph matches the height of your red one.)
But they call it an "integral" rather than an "antiderivative" - why?

In the case of motion with known velocity it is possible to express the distance travelled as a sum of distances covered over each of a number of short time intervals, and each of these can be approximated by an instantaneous velocity times the length of the time interval (see "differential approximation"). If the approximation can be made arbitrarily good by taking small enough time intervals, then we can express the distance as a limit of sums - which has the same form as our definition of a definite integral!

This idea is explored in more detail in the page on Accumulating Rates of Change, in the Gallery of Interactive Geometry, at the University of Minnesota's Geometry Center.

This relationship between derivatives and integrals is important enough to be known as the Fundamental Theorem of Calculus, and is illustrated on the web by a number of animations and java applets.

Many applications of antiderivatives are related to the fact that the "area-so-far" function under the graph of a positive function is an antiderivative for the given function.

By similar reasoning, you might agree that as a vessel is filled, the rate of change of volume with respect to depth is equal to the surface area. So the volume V(h) up to depth h is an antiderivative for the function S(h) which gives the surface area. (This is related to the issues addressed in the first lab in our Math1183 course)

### Finding Antiderivatives

One way to construct an antiderivative for a given function is to use a limit of sums, as described above for the distance problem, or as in the Riemann Sum method of approximating area under a curve. This would find the antiderivative by expressing it as a definite integral with the variable as right endpoint (sometimes called an "indefinite integral").

But it can be much easier than this - especially if we recognize the function whose antiderivative we want as being the result of a derivative we have already done.

Eg.   Since the derivative of x^2 is known to be 2x, we know at once that an antiderivative for 2x is given by x^2, and if we want an antiderivative for just x...
Well then (x^2)/2 would do!  (and of course so would (x^2)/2 + C, for any constant C)

Can you use the known rule for d/dx(x^3) to find an antiderivative for x^2?

In fact, it is far more common to use antiderivatives found this way to compute integrals than it is to go the other way and use integrals to compute antiderivatives.

Because of this, the set of rules and tricks (like dividing by 2 above) that are used to find antiderivatives this way are often referred to as "Techniques of Integration".

You might also check our 'raw list' (of links provided without comment) to see if there are any more examples there that we haven't yet included here.

If you have come across any good web-based illustrations of these and related concepts,