*You might also like to read the second part of this article.*

What is the integral of ? If you're up to speed with your calculus, you'll know the answer off by heart. It's

(1) |

(2) |

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 might better be written as

### The integral of *x*^{2}

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 -axis and the vertical line at by a sequence of rectangles. The rectangles are constructed on intervals on the -axis of length . The height of each rectangle is the maximum value of the curve on the corresponding interval. In the case of this maximum value occurs at the right hand end of each interval. For the th interval this right hand end point is and the corresponding value of the function is The area of the th rectangle is therefore

*from below*, using rectangles with height equal to the minimum value of the function on the corresponding interval. If we repeat this analysis we have the area as

### The integrals of *x*^{k} 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 and a positive real number . Fermat's method uses *rectangles of unequal width*. To start, fix an integer and a number with . Now consider rectangles whose vertical sides meet the -axis at points of the form for .

The graph of *f(x)=x ^{2}* with

*b*=0.8, and with

*x*. Use the sliders to change the value of

_{i}=r^{i}b*r*(and therefore the widths of the rectangles), and to change the value of

*n*to see the position of the

*n*th rectangle. Created with GeoGebra

Now, the -coordinate of the right hand end of the th rectangle is . The -coordinate of the left hand end of the th rectangle is . If we subtract the -coordinates of the ends of rectangles to get the width, we have . The height of the th rectangle is . The total area of these rectangles is then

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 th rectangle is , which tends to as . (Use the sliders in the figure above to see how the widths of the rectangles change as tends to 1, and how far to the left the rectangles extend as gets larger.) Therefore, the area under the curve is equal to

### 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 .

*You might also like to read the second part of this article.*

### 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.