How to calculate definite integrals of a function.
Consider the problem of finding the area ‘A’ between a curved line: y = f(x) and the ‘x’ axis, between x = a and x = b:
We can approximate the area by slicing it into rectangles and adding up the areas of the rectangles. In the diagram below, there are three rectangles of equal width. The height of each rectangle equals the height of the function graph at its midpoint, and the width of each rectangle = (b-a)/3:
The total area under the
curve from x = a to x = b can be approximated by: A = A1 + A2
Notice that the area
A1 is an overestimate of the area under the curve, A2 is
about right, and A3 is too small.
Thus this method may be accurate for some types of curves, but not for
others. It depends on their shape.
One way to improve
the accuracy for any shaped curve is to increase the number of rectangles. If
there are ‘n’ rectangles, then as ‘n’ increases, the width of each
For very thin
rectangles, areas at the top that don’t fit the curve become insignificant,
then the sum of the rectangles becomes very close to the area under the curved
Let be the area under the graph: y = f(x) from x = a to x = b. We can calculate the area by summing up the areas of the rectangular slices. If there are ‘n’ slices, then the total area is approximated by:
The true area is called the 'definite
integral' of the function, and can be written as the limit of this sum as
‘n’ approaches infinity, like this:
The 'Method' section below shows you how to use Maths Helper Plus to calculate the definite integral of a function by adding rectangles under its graph. This can be very accurate because the computer can add many thousands of tiny rectangles in a fraction of a second.
To explain how to use Maths Helper Plus to calculate a definite integral, we will use the function:
We will calculate the definite integral from x = -3 to x = 4.
Step 1 Start with an empty Maths Helper Plus document
If you have just launched the software then you already have an empty document, otherwise hold down ‘Ctrl’ while you briefly press the ‘N’ key.
Step 2 Graph your function
For more help on entering functions, see the 'Easy Start' tutorial in the Maths Helper Plus 'help'. To access the tutorial, click 'Help' on the Maths Helper Plus menu bar, then select: 'Tutorial...'
If the graph of the function does not appear, you need to adjust the
For more help on setting graph scales, click 'Help' on the Maths Helper Plus menu bar, then select 'Index'. Click the button at the top left corner of the help screen. Click 'Training', then 'Essential skills', then 'Graph scale magic'.
Step 3 Display the options dialog box and set options
Carefully point to the plotted function graph line with the mouse and double click the mouse. This will display the function options dialog box. Click to select the 'Integrals' tab at the top of the dialog box:
Step 4 Read the integral value from the table
The text view displays the table of calculated integrals. This is how the table looks for the example integral:
Simpson's rule was fastest to achieve four decimal place accuracy.
For the trapezoidal rule, four decimal place accuracy is achieved at n=300. This is proved because the last digit (7) does not change for n = 500.
For the midpoint method, n = 500 is required to achieve four decimal place accuracy.
NOTE: We could add another higher 'n' value to the list, say n = 700, to check out the accuracy of the last digit for the midpoint method, but since we already know this answer is correct from the other two columns, we don't need to do so.