Area of A Region

Apply the definite integral to find the area of a region under curve, and then use the GraphFunc utility online to confirm the result.  See the demo.

Let the nonnegative function given by y = f(x) represents a smooth curve on the closed interval [a, b].  The area of the region bounded by the curve of f(x), the x-axis, and the vertical lines x = a and x = b, as shown in Figure 1, is given by # Basic Properties of Definite Integrals

·         If f is defined at x = a, then ·         If f is integrable on [a,b], then  Figure 1.

·         If on [a, b], then ·         If f < 0 on [a, b], then Example 1.  Find the area of the region bounded by , y = 0, x = 0 and x = 2.  Use the GraphFunc utility to confirm the result.

Solution

The graph of is shown in Figure 2.    ( = 2.66666…) ## Figure 2

 Use the GraphFunc utility online to verify the above result as shown in the following steps:

• Begin by going to http://graph.seriesmathstudy.com/ (if this is the first time you load this page, you may have to wait for the applet to be loaded.  Also, your browser needs JRE 1.4x to run this applet.)
• Enter the expression x^2 at command line and click on the Graph It! button.
• Enter the limit values x = 0 and x = 2 into the text fields: .
• Click on the Find Area button to compute the area of the graph as shown in Figure 3. ## Figure 3

We see that the areas that GraphFunc computes and the one we derive are the same.

Example 2.  Find the area of the region bounded by , y = 0, x = 0 and x = 2.

Solution

The graph of is shown in Figure 4.    ( ) ## Figure 4

Use the software GraphFunc online to verify the above result as follows:

• Begin by going to http://graph.seriesmathstudy.com
• Enter the expression x*sin(x) at command line and click on the Graph It! button.
• Enter the limit values x = 0 and x = 2 into the text fields: .
• Click on the Find Area button to compute the area of the graph as illustrated in Figure 5. 