
What is the integral of

Pierre de Fermat, 1601-1665.
But can you prove that this is true? In this article we'll derive (1) from first principles, using an ingenious method devised by the mathematician Pierre de Fermat in the 17th century. In the second part of this article, we'll examine the surprising fact that, at a symbolic level, the answer to
The integral of x2
Let's start by deriving the integral

One approach is to use the above diagram, where we have approximated the area between the curve, the
The integrals of xk for k>2
It is clear that this method could, in principle, be extended to find

Isaac Newton, 1643-1727.
Before Isaac Newton's discovery of the fundamental theorem of calculus, which allows integrals to be evaluated in practice as anti-derivatives, resorting to such sums was the only way to calculate integrals as areas. Newton and Gottfried Wilhelm von Leibniz made very significant advances in the development of calculus as a systematic set of tools. This work began at least with Archimedes and has a continuous history (see reference [2] below). For example, work on the integration of rational functions was collected together by the mathematician G.H. Hardy in 1916 (see reference [3]), but it was only with the advent of computer algebra systems that an algorithm was developed which determined whether an elementary expression could be integrated symbolically at all (see reference [4]).
One of these incremental developments was discovered by Fermat who devised a method for calculating
First, choose a positive whole number
The graph of f(x)=x2 with b=0.8, and with xi=rib. Use the sliders to change the value of r (and therefore the widths of the rectangles), and to change the value of n to see the position of the nth rectangle. Created with GeoGebra
Now, the
The sum in the right hand side of this expression is a geometric progression, which we can evaluate using the standard formula

Gottfried Leibniz, 1646-1716.
Now notice that the more rectangles we use, the better our approximation of the area under the curve. Letting the widths of all the rectangles tend to zero, and therefore their number to infinity, will give us the area of the curve as a limit. The width of
Have a go yourself
If this article has inspired you to do some calculus, here are a couple of problems for you:
- Use the method described here and the result
to find the area under . - Use apply Fermat's method of unequal rectangles to calculate the area under
between and .
References
[1] W.G. Bickley, §1324. An adventure with limits, The Mathematical Gazette, 22(251):404-405, October 1936.
[2] C.H. Edwards, The Historical Development of the Calculus, Springer-Verlag, 1979.
[3] G.H. Hardy, The integration of functions of a single variable, Tracts in Mathematics and Mathematical Physics. Cambridge University Press, 1916.
[4] R.H. Risch, The problem of integration in finite terms, Transactions of the American Mathematical Society, 139:167-189, May 1969.

Chris Sangwin is a member of staff in the School of Mathematics at the University of Birmingham. He has written the popular mathematics books Mathematics Galore!, with Chris Budd, and How round is your circle? with John Bryant, and edited Euler's Elements of Algebra.
Comments
Formatting Error
There is an error in formatting halfway down the page
Thanks for poiting that out,…
Thanks for poiting that out, w ehave fixed it!