Areas under a hyperbola and the logarithm
The function has some remarkable properties if we look at the areas underneath it. This question of how much area is under a particular graph is one of the two central problems of calculus. But sometimes we can derive surprising relations without calculus, as we do here.
In this step, we
look at a lovely symmetry about areas under the hyperbola
introduce the logarithm function in an elementary way, without calculus
This is a more advanced step, but if you go carefully, then you can follow all the arguments, which are really quite spectacular in their consequences!
First let’s note that any point on the graph of the hyperbola, say where , determines a rectangle with the and axes. And the area of this rectangle is always exactly , since the base and height of this rectangle are and , which multiply to .
Q1 (E): What is the common area of such rectangles for the hyperbola ?
But other kinds of areas under this graph are also interesting, and exhibit an interesting property when we scale things.
Q2 (E): Explain why the two rectangular areas are equal.
A mixed dilation of the plane
Consider a transformation of the plane that sends a point to
This is a stretch in the horizontal direction by , and at the same time a shrink by in the vertical direction. So does not change the areas of rectangles. And since general regions are made up (approximately) of rectangles, we can reason that although will change the shape of regions, it will not change their area.
But what do we mean by area of a region? Although we have a clear understanding of areas of rectangles, the definition of “area” for a curved region is much more problematic logically. The 17th century mathematicians already realised this. In practical applied mathematics we just assume that the term consistently extends to more complicated curved regions, but in pure mathematics trying to pin the term down embroils us in all kinds of subtle issues.
So let’s adopt a pragmatic attitude, not worry about the logical difficulties, and so assume that the word “area” really makes sense even for regions bounded by graphs like .
The action of on the graph of
Let’s observe that points on the -axis get moved by to other points on the -axis, so that for example the interval will be sent to the interval . The same holds for points on the -axis.
Now if is a point on the graph of , then
is still a point on the graph. This means that the entire graph of is mapped to itself by .
And now since our graph is actually preserved by , the area under it from to must be the same as the area under it from to . This is an elementary but remarkable observation!! We hope you have been able to follow the argument. Well done if you have!
An unbounded area!
If we apply again, then we see that the area under the graph from to is also exactly the same as the previous two areas! And similarly the area from to is the same, and from to is the same.
As we continue this, we conclude that the total area under the curve from onwards exceeds any known bound, or is unbounded. So we can state an important theoretical result: The area under the graph from to is unbounded.
Introducing , as areas under
If you have followed this argument so far, we can go just a little bit further to obtain an even more remarkable relation. Suppose we consider the areas under over the three intervals , and . Let’s call these areas , and .
The same argument we have already used shows that is the same as , since this time we dilate and shrink by and . So then
The same argument applies for more general numbers:
In fact this is the same rule as satisfied by the function, often called the natural logarithm: namely
It turns out that this is more than just a coincidence, in fact
In fact in some treatments of calculus, the function is defined precisely this way — as the area under from to . So a large part of the calculus that deals with logarithm and exponential functions ultimately hinges on the remarkable hyperbola and its geometrical properties!
Q3 (E): What is the value of ?
Q4 (C): (for discussion) Is it possible to assign a certain value for ? What should this value be?
Q5 (C): (for discussion) how do you pronounce the function ?
A1. Since the typical point on the hyperbola is the area of rectangles is
A2. The second rectangle has twice the base of the first, but half the height. So the areas must be equal!
A3. As the number is defined as the area between the x axis and the graph of from to , we deduce that
A4. First of all for the value of needs to be negative. The reason is a little bit subtle: actually we want to consider ‘signed areas’ and since we are going backwards from to the value is deemed to be negative. And what happens as gets closer and closer to , but from the positive side? Well then the area under the graph gets to be a larger and larger negative number.
And why is that? It is because the curve is symmetric if we interchange and , so the two areas, blue and red, shown here had better be the same – and our earlier argument showed that the blue area was unbounded. So the red one must also be unbounded!
A5. Norman says is “lon”. Daniel says “natural logarithm”.
© UNSW Australia 2015