Plotting points on a calculator to find the area under curves/chapter 13
I was browsing through a Scientific American "Computer Recreations" article in which it looked like they were filling in squares on a computer screen. I asked myself "Could I do that on our programmable graphics calculator?" After a couple of hours of trying things and making mistakes (I was never very good at programming), I was able to plot points on the screen to make a 1x1 square starting on the left, below. I counted the number of points plotted, and this number was a measure of the area of the square. I wrote the program so it was possible to change the upper level of the points that are plotted, because the next thing I would wanted to do was to plot the points under a parabola, like x^{2}, or x^{3}, or a quartercircle, a sine wave, or whatever. Very exciting! The figure on the right was the area under a parabola, from x=0, to x=1; as predicted, this area came out to be very close to 1/3.
My progam to plot the points to make a square:


Running this program fills in a 1x1 square; the number of dots varies with the calculator or computer. On my calculator I got 400 points. To change the program to plot the points under a parabola y = x^{2} from 0 to 1, replace the 1 in line 5 with x^{2}. In the basic program change statement 20 from 81 to x^{2}. I got the picture on the right with N=133. So the area under the curve y = x^{2} from 0 to 1 was 133/400, very, very close to 1/3 = 133/399, which is the integral of x^{2}, from 0 to 1!! Needless to say, this was very exciting! I showed it to everyone teachers, students, and parents. [N.B. you might have to adjust the programs for your computer or calculator].
The following is a flowchart to help write the programs above.
Well now, if I don't have a calculator or computer can I still do this? SURE! I started doing this on 1/10" graph paper.
added the colors and some lines to make things more readable)
Matt, a 7 year old, counted the squares under y = x2 from x = 0 to x = 1.
He counted 32 squares, and 32/100 of the 1 x 1 blue square below as
the area. Byron, a 10 year old recently doing this, counted 42 squares and I
told him to go back and do it again and be more careful. He too came up with 32.
I asked Matt what simple fraction was close to this.
He said 1/3; I wrote this as 1/3*1*1^{2} or 1/3*1^{3}.
From 0 to 2 he correctly predicted 1/3 of the 2x2^{2} rectangle
or 1/3*2^{3} (the green rectangle
above) as the area under the curve y = x^{2} from
x = 0 to x = 2.
Sean counted approximately 279 squares in going from 0 to 2, but was satisfied
this was close enough. He figured there are 20*40 = 800 little squares in the
2x4 green rectangle and the area should be 1/3 of 800 or 266.66... which is
close to 279. The notation I used on his graph I now have changed to and use
A01 (x^{2}) = 1/3*1^{3}.
Then the area from 02 he wrote as A02 (x^{2})
= 1/3*2^{3}. Writing the area this way
instead of 1 and 8, makes it possible to see a pattern. I always want my
students to look for patterns. Writing the answers different ways often helps to
see the pattern.
What would you predict would be the area from 03? 04? 0n?
How about finding the area under the curve y = x^{3} from
01? 02? 0n?... Look for patterns!
See theintegral