 Fermat also considered the problem of finding the area under a curve. Today we'd say he found the area under the curve y equals 1 over x to the n over the interval from 1 to infinity. Now, Fermat's original work used some rather abstruse geometric arguments. We won't try to reproduce it here. Instead, we'll focus on the main points. Fermat knew the following, which we state in modern terms. But a, a r, a r squared, a r cubed, and so on be a geometric sequence with common ratio r. Then a over 1 minus r is the sum of the infinite series. The important thing here is that if we can partition a region into areas in geometric progression, we can find the sum of all of the areas. So Fermat considered two types of regions. If m and n are non-zero whole numbers, we can consider generalized parabolas. These are curves satisfying x to the m equals p y to the n over the interval between 0 and b, and generalized hyperboloids curve satisfying x to the m y to the n equals p over the open interval from 1 to infinity. So Fermat illustrated his method using the hyperboloid x squared y equals 1. So let's take r greater than 1. For r greater than 1, we can partition the interval from 1 to infinity at the points 1, r, r squared, r cubed, and so on. And then we can form rectangles in each of these regions. So for this first rectangle, the partition point is x equals 1, and the height, 1 over 1 squared, or 1. The width is r minus 1, and so the area, 1 times r minus 1. So we want to find the total areas of all the rectangles, so we'll start off with the area of that first rectangle. How about the second rectangle? So for the second rectangle, our partition point is at x equals r, and the height is 1 over r squared. The width is r squared minus r, and so the area is 1 over r squared times r squared minus r. But we can do a little algebraic simplification and get the area of the second rectangle. How about the third rectangle? The partition point is x equals r squared, and the height is 1 over r to the fourth. The width is r cubed minus r squared. The area, 1 over r to the fourth times r cubed minus r squared, which simplifies giving us the area of the third rectangle. Well, as they say, fourth times the charm. If we take a look at our expression for the total area, we see that it's a geometric series with first term r minus 1 and common ratio, 1 over r. But there's an app for that. Well, there's a theorem at least. And our theorem says the sum of the series is the first term, r minus 1, over 1 minus the common ratio, 1 over r. And if we do a little simplification. Now, notice that we haven't actually found the area under the curve. We found the area of this, we'll call it a staircase region. But notice that as r gets close to 1, the rectangles get narrower and narrower. And so the sum of all of these areas gets closer and closer to the actual area under the curve. So in modern terms, as r gets close to 1, our area gets close to 1 as well. Now, Fermat could have done y equals x squared over the interval between 0 and 1 in exactly the same way. Instead, he took a different approach. Suppose we take r less than 1. To produce a more recognizable geometric series, Fermat partitioned the vertical axis at 1, r squared, r to the fourth, r to the sixth, and so on. And again, he computed the areas. The first rectangle has height 1 minus r squared with square root of 1, so its area will be 1 minus r squared. The second rectangle has height r squared minus r to the fourth, and the width will be the square root of r squared, or just r. So its area will be, which we can factor, the third rectangle has height, width, and area, which we can factor as, and the fourth rectangle. And so once again, our areas form a geometric sequence, and the total area is a geometric series. So we have the first term of the series, 1 minus r squared, and the common ratio is r cubed. And so the sum of the series, 1 minus r squared over 1 minus r cubed, which simplifies to, and as r gets close to 1, our area gets close to 2 thirds the area. And since this entire region fits in a rectangle that's 1 wide by 1 high, we can find the area under the curve, 1 minus 2 thirds, or 1 third. It's worth pointing out that unlike the method of tangents, which can be explained purely in terms of the theory of equations, Fermat's approach to areas does require us to introduce a limit concept. So this raises an important question. Did Fermat invent integration? Fermat's method can be used to find the areas under curves we'd express as y equals x to the n for any rational value of n, except for n equals minus 1. And in fact, we can turn Fermat's work into a general rule. The integral from 0 to 1 of x to the n, as long as n is greater than 0, 1 over n plus 1. Meanwhile, if n is greater than 1, the integral from 1 to infinity of 1 over x to the n is 1 over n minus 1. And so we might ask the question, did Fermat invent calculus?