Area between two curves
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How to calculate the area between two graph curves.

This topic is part of the TCS FREE high school mathematics 'How-to Library', and will help you to calculate the area between two graph curves.
(See the index page for a list of all available topics in the library.) To make best use of this topic, you need to download the Maths Helper Plus software. Click here for instructions.

Theory:

To find the area between two intersecting curves that only intersect at two points, we first find the ‘x’ coordinates of the two intersection points: x = a and x = b. Definite integrals give us the area under each curve from x = a to b, then we subtract the two areas to obtain the area between the curves. In the diagram below, the area between the two graphs is shaded:

The 'method' section below will use the graphs in the diagram above as examples. Use the same steps to find the areas between other graphs.

Download the free support file... We have created a Maths Helper Plus document containing the completed example from this topic. You can use this to practice the steps described below, and as a starting point for solving your own problems.

File name:  'Area between curves.mhp'   File size: 3kb
Click here to download the file.

If you choose 'Open this file from its current location', then Maths Helper Plus should open the document immediately. If not, try the other option: 'Save this file to disk', then run Maths Helper Plus and choose the 'Open' command from the 'File' menu. Locate the saved file and open it. If you do not yet have Maths Helper Plus installed on your computer, click here for instructions.

Method:

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 the two functions

1. Press the F3 key to activate the 'input box' for typing (see below):

2. Type the first function into the input box:

             

3. Press Enter to complete the entry

4. Type the second function into the input box:

              

5. Press Enter to complete the entry

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 required area between the function graphs is not completely visible,  you need to adjust the graph scale.  First reduce the graph scale. Press the F10 key enough times until the main parts of the graph are visible. To enlarge the graph, hold down 'Ctrl' while you press F10.

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  Find  the intersection points between the graphs

Click the    toolbar button to select the intersection tool. 

Click the mouse cursor at the intersection points of the two curves.

Record the ‘x’ coordinates of the intersection points:

   x (left) = ‘a’ =   ___________

 x (right) = ‘b’ =  ___________

Click ‘Cancel’ on the intersection tool dialog to cancel the intersection tool.

 Step 4  Find the area between the graphs and the 'x' axis from 'a' to 'b'

Carefully point to one of the function curves with the mouse pointer. Double click to display the options dialog box for that function. (This may work better on parts of the graph that are not so steep.)

Click the 'Integrals' tab at the top of the dialog box. See below:

Click on the ‘Limits of integration’ edit box, and type the the ‘a’ and ‘b’ values (from step 3 above) with square brackets, like this: [-1,2]

Click on the ‘Number of intervals’ edit box. Type : 100. (The larger this number the greater the accuracy, but if it is too big the calculations may be slow. This must be an even number to use Simpson’s rule.)

Select ‘Shade areas on graph’. Also select ‘Simpson’s rule’. ‘Calculation mode’ should be ‘definite integral’. Click the ‘OK’ button to close the dialog box.

The area under this function will be shaded on the graph, and the definite integral will be displayed on the text view.

For the example equation: y = x + 3, the definite integral is displayed as follows:

Definite integral from x = -1 to x = 2

n    Simpson

100  10.5

The absolute (positive) value of the definite integral equals the area 

                             of the first function, = _________________

For the example equation y = x + 3, the area between this graph and the 'x' axis from x=-1 to x=2 is 10.5

 

Repeat step 4 for the other plotted function.

Do exactly the same things as for the first function, except that you need to change the shading colour. Click the 'Shade colour...' button and select pale blue. The area between the two curves will now have its own colour.

The absolute (positive) value of the definite integral equals the area 

                             of the second function, = _________________

For the example equation y = x²+1, the area between this graph and the 'x' axis from x=-1 to x=2 is 6

 Step 5  Calculate the area between the curves

 Calculate the area between the curves by subtracting the smaller area from the larger area.

 Area between curves =  ___________  -  ____________

                                      =  ___________

For the example equations , the area between this graphs = 10.5 - 6 = 4.5

 

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