We are going to focus on this method by looking at an example. Our initial objective is to find the area of the plane region bounded by the X-axis, the graph of the function y = x² and the straight lines x = 0 and x = b, where b is any real number. More specifically, what we want to find is the shaded area of the following graph:

Try different values of b.

The method of exhaustion, used by Archimedes to solve this very problem, consists of the following (expressed using more modern terms):

For each natural number n we divide the segment [0,b] into n equal parts of length b/n. Each of these parts forms the base of a rectangle whose height is equal to the highest y-coordinate (superior rectangle, above the curve or circumscribed).

Complete the following activities in the window below:

1.- Change the values of n and b to observe what is mentioned in the paragraph above. If n is too big the base of the rectangles will be too small to see them clearly. If this is the case, change the scale or the position of the axes to see them more clearly.

2.- For each value of n we shall call the sum of the areas of all the superior rectangles S_n (or upper sum). What do you notice about S_n compared to the value of the area we are looking for? Is it true for any value of n?

3.- Let n₁ and n₂ be two possible values of the parameter n, and n₁ < n₂, What can be said about the respective values of  S_n1 and S_n2?

4.- Try to write a theoretical statement which summarises the results of the two questions above.

Now let's repeat the process with rectangles whose height is the lowest possible y-coordinate on the graph (inferior rectangle, below the curve or inscribed).

Complete the activities listed below:

1.- For each value of n we shall call the sum of the areas of all the inferior rectangles s_n (or lower sum). What do you notice about s_n compared to the value of the area we are looking for? Is it true for any value of n?

2.- Let n₁ and n₂ be two possible values of the parameter n, and n₁ < n₂. What can be said about the respective values of s_n1 and s_n2?

3.- Try to write a theoretical statement which summarises the results of the two questions above.

Now let's look at the two situations simultaneously.

Complete the activities listed below:

1.- What is the relation between any "upper sum" S_n and any "lower sum" s_m?

2.- If A is the area we want to find, what is the relation between A, S_n and s_m? 

3.- What can we say about S_n - s_n  when n tends to infinity?

4.- Find the original area for different values of b and to an error of less than a tenth.