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Area Under a Curve Using an Infinite Number of Rectangles

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					    Area Under a Curve Using
      an Infinite Number of
            Rectangles




A better approximation of the area under a
curve can be found by dividing the area under
the curve into an infinite number of
rectangles.

Ex. Find the area under the curve bounded
    by:
     y = 5 - x2, x = 0, x = 2 and the x -axis.




                                                 1
Step 1: Divide the interval [0, 2] into n
        intervals (rectangles) of equal
        width. List the endpoints.




                                            Pull
                                            Pull
                                            Pull
                                                   Endpoints




Step 2: Find an expression for the right
        endpoints.


Right Endpoints:
    0 + 2/n, 0+ 2(2/n) + 0 + 3(2/n) + ...
  = 2/n, 4/n, 6/n, ...
  or
  = 2i/n, where i = the number of
           rectangles




                                                       2
Step 3: Use the endpoints to determine
        the heights of each rectangle.

heights = f(2i/n)
        = f(endpoints used)



(substitute 2i/n into the function.)




 Step 4: Write a series in sigma notation
         that describes the area of each
         rectangle.




                      (width X height)




                                            3
              Step 5: Evaluate the series using the
                      appropriate formulas.




By Integration:




                                                      4
5
Assignment
Determine the area under the curve of the following
regions using:
1. exactly 5 rectangles and left endpoints
2. exactly 5 rectangles and right endpoints
3. an infinite number of rectangles
4. integration

Question 1:       y = 2x2-x+1, [0, 4] and x-axis

Questions 2: y = 16-x2, [0, 4] and x-axis

Question 3:       y = (x-4)2, [4, 6] and x-axis


                                                      6

				
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posted:1/26/2011
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