Approximations to Integrals
- Author:
- Andy Wain
Area by Rectangles
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.
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)
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:
Type in the rule, (type: 5x - x ^ 2)
a. Make the lower bound 0 and the upper bound 5 and divisions 5.
b. Complete:
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:
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).
- Now change the value of n and complete the table:
| n | Width | Upper Rectangles | Lower Rectangles | Actual Area |
| 4 | 1 | |||
| 8 | ||||
| 20 | ||||
| 40 | ||||
| 100 |
| Rule | Upper Rectangles | Lower Rectangles |
| n | width | Upper | Lower | Left | Right | Actual |
| 5 | ||||||
| 10 | ||||||
| 20 | ||||||
| 50 | ||||||
| 100 |
| n | width | Upper | Lower | Left | Right | Actual |
| 5 | ||||||
| 10 | ||||||
| 20 | ||||||
| 50 | ||||||
| 100 |
| n | width | Upper | Lower | Left | Right | Actual |
| 5 | ||||||
| 10 | ||||||
| 20 | ||||||
| 50 | ||||||
| 100 |
-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).