# A question about Fermat's method of calculating areas under curves

1. Nov 7, 2011

### murshid_islam

I am currently reading the book "e: The Story of a Number" by Eli Maor. And I got stuck at something. In chapter 7 of the book, the author described the method Fermat used to calculate areas under curves of the form $y = x^n$, where n is a positive integer. I am quoting the relevant bit here (sorry, I can't show the figure, but from the description, you can easily receate it):

Now I can't get to the final formula. The areas of each rectangle I found are $a^{n+1}, (ar)^{n+1}, (ar^{2})^{n+1},$ and so on. Their sum,

$$A_{r} = a^{n+1} + (ar)^{n+1} + (ar^{2})^{n+1} + \cdots$$
$$= a^{n+1}\left(1 + r^{n+1} + r^{2(n+1)} + \cdots \right)$$
$$= \frac{a^{n+1}}{1 - r^{n+1}}$$

Where am I getting wrong?
.

Last edited: Nov 7, 2011
2. Nov 7, 2011

### mathman

You are using the ordinates to get the heights of the rectangles. I think the formula is based on using the averages of the adjacent ordinates to get the rectangle heights,

3. Nov 8, 2011

### murshid_islam

Thanks, but that was not it. I've just figured out my mistake. I got the areas of the rectangles wrong. The sum of the areas would be,

$$A_r = (a - ar)a^n + (ar - ar^2)(ar)^n + (ar^2 - ar^3)(ar^2)^n + \cdots$$
$$A_r = a^{n+1}(1 - r) \left(1 + r^{n+1} + r^{2(n+1)} + \cdots \right)$$
$$A_r = \frac{a^{n+1}(1 - r)}{1 - r^{n+1}}$$

4. Nov 8, 2011

### Stephen Tashi

I don't understand how you got those areas. Does the base of the first rectangle have length = a or does it have length = (a - ar)?

5. Nov 8, 2011

### murshid_islam

Yes, that's the mistake I made. I posted the correct calculation in this post: https://www.physicsforums.com/showpost.php?p=3604692&postcount=3"

The bases are (a - ar), (ar - ar2), (ar2 - ar3), and so on.
.

Last edited by a moderator: Apr 26, 2017