[ Home ] [ Up ]
 Welcome to the Teachers' Choice Software web site. Our FREE on line mathematics 'How-To' library is open 24 hours a day.  The ever-growing library now has 71 helpful topics, and every topic includes a free download!

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:)

 Drag on the title bar at the top of the angle tool dialog box to move it. Check the 'Snap to grid' option to locate the limits of integration precisely at grid snap positions. Click the 'Clear' button to remove temporary shaded areas from the graph view. Click the 'Cancel' button to stop using the integral tool.

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.

 In the diagram (right), the definite integral of the function (plotted as the blue curve) from x = 6 to x = 9 is equal to the yellow shaded area:

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:

 In the diagram (right), the definite integral is found by subtracting the shaded area below the 'x' axis from the shaded area above the 'x' axis. Because more of the area lies below than above, the definite integral from x = 3 to x = 6 has a negative value.

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:

1. If necessary, set the graph scales so that the part of the function that you want the definite integral of is displayed in the graph area of the graph view.

2. Select the integral tool, then move the integral tool cursor: onto the graph plot at x = a, the lower (left hand) limit of the integral.

3. Click and hold the left mouse button while you drag the mouse to x = b, the upper (right hand) limit of the integral.

4. Release the mouse button.

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.

 Simpson's rule is generally more accurate. This rule fits a parabola to the curve for each slice: The Trapezoidal rule uses a straight line to approximate the curve:

The width of slices used by the integral tool to calculate an integral depends on the 'snap to grid' check box setting.

 If the 'snap to grid' check box is checked, then each slice is the width of the grid snap squares. (These are user definable.) If the 'snap to grid' check box is not checked, then the slices are one screen pixel wide.

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.