# Approximations to Integrals

Author:
Andy Wain

## 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.
1. Now change the value of n and complete the table:
 n Width Upper Rectangles Lower Rectangles Actual Area 4 1 8 20 40 100
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)
 Rule Upper Rectangles Lower Rectangles
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 10 20 50 100
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 50 100
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 20 50 100
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).