Area under a curve
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How to calculate areas under a function graph.
This topic is part of the TCS FREE high school mathematics 'How-to Library', and will help you to calculate areas under a function graph.
(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:
The definite integral can be used to find the area between a graph curve and the ‘x’ axis, between two given ‘x’ values. This area is called the ‘area under the curve’ regardless of whether it is above or below the ‘x’ axis.
When the curve is above the ‘x’ axis, the area is the same as the definite integral ...
but when the graph line is below the ‘x’ axis, the definite integral is negative. The area is then given by:
Sometimes part of the graph is above the ‘x’ axis and part is below, then it is necessary to calculate several integrals. When the area of each part is found, the total area can be found by adding the parts.
For example, to find the area between the graph of: y = x² - x - 2 and the ‘x’ axis, from x = -2 to x = 3, we need to calculate three separate integrals:
The zeros of the function f(x) that lie between -2 and 3 form the boundaries of the separate area segments.
In this case there are zeros at x = -1 and x = 2, (see graph above) and so three separate areas must be found: A1, A2 and A3 as follows:
So the total shaded area between the function and the graph from x = -2 to x = 3 is given by:
A = A1 + A2 + A3
Maths Helper Plus can graph the function, locate the zeros and calculate the definite integrals. Follow the steps below to find the area under a curve as described in the example above.
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 under curve.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:
To explain how to use Maths Helper Plus to calculate between a function graph and the 'x' axis, we will use the function:
y = x² - x - 2
We will calculate the area between the curve and the 'x' axis from x = -2 to x = 3.
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
1. Press the F3 key to activate the 'input box' for typing (see below):
2. Type the function into the input box:
3. 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 graph of the function does not appear, 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 Locate zeros on the graph
If the graph crosses the 'x' axis between the limits x = a and x = b where we are calculating areas, then we have to calculate separate areas and add them.
For the example: We want to find the area under the curve between x = -2 and x = 3. The curve crosses the ‘x’ axis between these ‘x’ values, so we use the ‘intersection tool’ to locate the zeros.
Click the
toolbar button to select the intersection tool. Now click the mouse cursor on the points where the graph cuts the ‘x’ axis.
In each case, read and record the ‘x’ coordinate of the intersection point from the dialog box.
The example function has zeros at: x = -1 and x = 2
Cancel the intersection tool by clicking the 'Cancel' button in the dialog box.
Step 4 Identify the integral boundaries
The areas will be calculated by finding definite integrals. Identify the number of integrals that need to be found. Write down the boundary ‘x’ values for each of these integrals.
For the example, there are three integrals to be found:
Integral 1: from -2 to -1.
Integral 2: from -1 to 2.
Integral 3: from 2 to 3.
Step 5 Calculate the definite integrals
The areas will be calculated by finding definite integrals. Identify the number of integrals that need to be found.
Carefully point to the function curve with the mouse pointer then double click with the mouse to display the options dialog for the function. (This works better on not-so-steep parts of the graph.) Select the ‘Integrals’ tab:
Click on the ‘Limits of integration’ edit box. Type the integral boundaries you calculated in step 4 above, using square brackets for each separate integral, like this: [-2,-1] [-1,2] [2,3]
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 areas will be shaded on the graph, and the integrals will be displayed on the text view.
Step 4 Read the integral values from the table and calculate the required area
The text view displays the table of calculated integrals. This is how the table looks for the example:
Definite integral from x = -2 to x = -1
n Simpson
100 1.83333
Definite integral from x = -1 to x = 2
n Simpson
100 -4.5
Definite integral from x = 2 to x = 3
n Simpson
100 1.83333
Add the absolute value (take each number as positive) of the definite integrals to calculate the total required area. Thus for the example, the total area is:
Total area = 1.83333 + 4.5 + 1.83333
= 8.16666