# 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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