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 ofis 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 ofis 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.

 

Figure 5

 

 

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