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 xaxis, 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 

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:
Figure 5

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