THE PROBLEM OF FINDING AREAS
1. THE AREA UNDER A CURVE

Given a plane region, its area can be worked out by using inscribed or circumscribed polygons in or around the shape. This is called the method of exhaustion and was used by the ancient Greeks.

We are interested in finding the yellow area bounded by a blue line (which looks like a trapezium), a curve (the graph of a continuous function), the X-axis and the vertical lines x=a and x=b, where a and b are two real values. Archimedes (287-212 B.C.) was successful in calculating the area enclosed by a parabolic line.
1.- Look carefully at the following trapezium-like shape. If you had to choose a polygon to help you to work out the orange area which would you choose? Why?

Move the step control one unit up to see our suggestion.

2.- Analyse the advantages and disadvantages of choosing a rectangle to help you to work out the area. Compare your results to your answer in activity 1.

3.- Once you've taken this first step what would you do to get a closer approximation to the area?

Move the step parameter up unit by unit to see our solution

The more rectangles you use the better approximation of the area you will get. This method is as follows:

  1. Subdivide the interval [a,b] into subintervals which will form the bases of the rectangles.
  2. By subdividing the interval you will get a finite set of ordered real numbers {x0, x1, x2, x3...,xn} known as the partition of segment [a,b].
  3. Each partition made up of n+1 points determines n rectangles. In our examples the bases are all the same length but this is not necessary in general.
  4. The area is calculated of all the inscribed rectangles inside the trapezium-like shape and they are added together to find the total area. 
2. APPROXIMATING THROUGH UNDER ESTIMATION 
4.- Work out the grey area for each partition on the interval [a,b]. Check your answers on the right of the window.

The blue 'C' control will help you find the heights of each rectangle.

Change the solution parameter to 1 to check the solutions for each step.

5.- How has the height of the rectangles been chosen on each of the subintervals of the partition? Is the same criteria used in each region?

6.- Here is another trapezium-like shape. Repeat the process and draw some conclusions.

 

The following expression is known as the Riemann lower sum of f for partition P {x0, x1, x2, x3...,xn} 

where mi is the minimum value of function f on the subinterval [xi-1,xi].

3. OTHER POSSIBLE APPROXIMATIONS

In activity 2 the approximation we used gave us an area which was less than that of the trapezium-like shape. However, this is not necessarily the most accurate approximation.
 

7.- Deduce which other possibilities there are to use the same technique to find the yellow area using rectangles or any other more suitable shapes. Analyse the advantages and disadvantages of using each method.

Change the option parameter to see two new suggestions.

The step parameter allows us to obtain successive partitions.

The following expression is known as the Riemann upper sum of f for partition P {x0, x1, x2, x3...,xn} 

where Mi is the maximum value of function f on the subinterval [xi-1,xi].
8.- Find the area of each grey shape for each partition.

9.- Write the general formula to find an area using the trapezium rule.

10.- How many times would we have to repeat the process to find the shaded area?

11.- How do the grey areas change as the partitions become narrower?

12.- If the rectangle bases get smaller and smaller how do the upper and lower Reimann sums change?

Enrique Martínez Arcos
Spanish Ministry of Education. Year 2001

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