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# 2.5 Area Under the Curve

The following argument is due to Pierre de Fermat (1601-1665) [19, pages 219-222]. Later we will use Fermat's method to find the area under the curve for all in .

Let be a real number with , and let be the set of points in such that and . I want to find the area of .

Let be a positive integer. Note that since , we have

Let be the box

Thus the upper left corner of lies on the curve .

To simplify the notation, I will write

Then

and

Hence

Observe that we have here a finite geometric series, so

 (2.30) (2.31)

Now
 (2.32)

so
 (2.33)

Let be the box

so that the upper right corner of lies on the curve and lies underneath the curve . Then

Hence,

Since

we have

i.e.,

By combining this result with (2.33), we get

Since , we can rewrite this as
 (2.34)

2.35   Exercise. What do you think the area of should be? Explain your answer. If you have no ideas, take in (2.34), take large values of , and by using a calculator, estimate to three or four decimal places of accuracy.

2.36   Exercise. A Let be a real number with , and let be a positive integer. Then

Let be the set of points such that and . Draw a sketch of , and show that

The calculation of is very similar to the calculation of .

What do you think the area of should be?

2.37   Exercise. Using the inequalities (2.6), and the results of Bernoulli's table in section 2.2, try to guess what the area of is for an arbitrary positive integer . Explain the basis for your guess. ( The correct formula for for positive integers was stated by Bonaventura Cavalieri in 1647[6, 122 ff]. Cavalieri also found a method for computing general positive integer power sums.)

Next: 2.6 Area of a Up: 2. Some Area Calculations Previous: 2.4 Finite Geometric Series   Index
Ray Mayer 2007-09-07