The integral tool makes it easy to find the definite integral of a y=f(x) type function from x=a to x=b.
Select the integral tool by clicking the button on the graph tools tool bar, or by selecting the 'Integral' command from the 'Tools' menu.
When selected, the integral tool cursor: is displayed on the graph view, and the integral tool dialog box is displayed. (See below:)
The definite integral of a function: y = f(x) from x = a to x = b is equal to the area between the function curve and the 'x' axis from x = a to 'x' = b.
Where the curve lies below the 'x' axis, the definite integral is negative. This means that if part of the area is above and part below the 'x' axis, the definite integral equals the difference between the two. If more of the area is below the 'x' axis than above, the total result will be negative:
Check the area check box to calculate the total shaded area. If this check box is checked, areas under the 'x' axis are counted as positive and simply added to the total.
How to use the integral tool:
To find the definite integral of a function y = f(x) from x=a to x=b:
This diagram explains the steps described above:
The definite integral is calculated by two different techniques and displayed on the integral tool dialog box. (See below.)
How the integral tool calculates area under a curve:
Two different methods are used to calculate the definite integral with the integral tool: Simpson's rule and the Trapezoidal rule.
In each case, the shaded area is divided up into many thin slices between the graph curve and the 'x' axis. The areas of the slices are found and added together to give the definite integral.
The difference between the two methods used in in how they approximate the part of each slice that lies along the function curve.
The width of slices used by the integral tool to calculate an integral depends on the 'snap to grid' check box setting.
All slices are the same width, so the number of slices depends on the total width from x=a to x=b. The number of slices is reported on the integral tool dialog box as 'n = 50' etc.
To study the Trapezoidal and Simpson's rule, you can set the grid snap square size to a large value and check the 'snap to grid' check box. The shading of the area is according to the Trapezoidal rule, so the error between the straight line approximation and the graph curve can be easily seen for 'bendy' functions.