Approximations to Integrals
Upper and Lower Rectangles
Approximate Area under a curve
Worksheet for the above Applets
Estimating The Area Under a Curve
(You need a computer that has GeoGebra loaded)
1.
Open up the files from above
a. Area by rectangles.ggb
b. upper_lower_rectangles.ggb
2.
Using the file, upper_lower_rectangles.ggb, type in the box for the rule, x2 (type: x ^ 2)
a. Make the lower bound 0 (use thescroll bar up the top left).
b. Make the upper bound 4.
c. Make the number of divisions, n, 4 (this will make rectangles of width 1 unit).
d. Click on the Green checkbox (Show Upper Rectangles). Upper rectangles are made by using the upper (higher) y value for each division.
f. Calculate the area of each rectangle (A = L x W).
g. What is the total area?
h. Is this greater or less than the actual area under the curve from 0 to 4?
i. Using another colour and the applet repeat for lower rectangles. (Lower rectangles are made by using the lower y value for each division
j. The actual area will be between these two answers. Calculate it by using calculus.
- Now change the value of n and complete the table:
|
n
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Width
|
Upper Rectangles
|
Lower Rectangles
|
Actual Area
|
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4
|
1
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8
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20
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|
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40
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100
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l. What do you notices as the number of divisions increases?
3. Repeat the process for the following functions:
a.
b. (use 3 divisions initially, then 6, 15, 30, 100)
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Rule
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Upper Rectangles
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Lower Rectangles
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In Summary:
The area under a curve can be estimated by using _______ and _______ rectangles. As
the _______ of the rectangles ___________ the approximation is __________.
4.
Using the file, Areas by rectangles.ggb , type in the rule, (type: 25-x ^ 2)
a. Make the lower bound 0 and the upper bound 5 and divisions 5.
b. Complete:
|
n
|
width
|
Upper
|
Lower
|
Left
|
Right
|
Actual
|
|
5
|
|
|
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|
|
|
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10
|
|
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20
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|
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50
|
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100
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Type in the rule, (type: 5x - x ^ 2)
a. Make the lower bound 0 and the upper bound 5 and divisions 5.
b. Complete:
|
n
|
width
|
Upper
|
Lower
|
Left
|
Right
|
Actual
|
|
5
|
|
|
|
|
|
|
|
10
|
|
|
|
|
|
|
|
20
|
|
|
|
|
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50
|
|
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100
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c. Are the Upper Rectangles always equal to the Right rectangles?
d. Explain the difference between Upper,Lower, Left and Right rectangles.
5. Using the file, Areas by rectangles.ggb , type in the rule, (type: e ^ (0.5x) )
a. Make the lower bound 0 and the upperbound 5 and divisions 5.
b. Complete:
|
n
|
width
|
Upper
|
Lower
|
Left
|
Right
|
Actual
|
|
5
|
|
|
|
|
|
|
|
10
|
|
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|
|
|
|
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20
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|
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50
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100
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c. What is the area enclosed by , the 
-axis and the lines 
and 
?(width 1 unit)
i. Counting Squares; ( _____ sq. unit)
ii. Left – endpoint rectangles; ( _____ sq. unit)
iii. Right – endpoint rectangles;( _____ sq. unit)
iv. Calculus ( _____ sq. unit).