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 x2,
or a quarter-circle, 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.
progam to plot the points to make a square:
- On the calculator
- 1)Window -2.35, 2.35, 1, -1.55, 1.55, 1
- 2) 0 -> x (sets left side)
- 3) 0 -> N (counts points in both cases)
- 4) Lbl 3
- 5) 1 -> y (sets top of figure)
- 6) Lbl 4
- 7) Plot x,y (plots the point)
- 8) 1+N -> N
- 9) y-.07 -> y (moves next pt. down)
- 10) y >or= 0 then goto 4 (sets bottom)
- 11) x+.07-> x (moves next pt. to rt)
- 12) x <or= 1 then goto 3 (sets rt end)
- 13) "N = ": N Disp (displays no. of points)
- in Basic
- 10 FOR X=0 TO 9 STEP .1
- 15 N=0
- 20 FOR Y=0 TO 81
- 30 T=150 + 9*X
- 40 V=150-Y (30
and 40 adjust for
the way pts are plotted on the
- 50 PSET (T,V)
- 60 N = N+1
- 90 NEXT Y (moves next point up screen)
- 100 NEXT X (moves next point to right)
- 110 PRINT N
- 120 STOP
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 = x2 from
0 to 1, replace the 1 in line 5 with x2. In the basic program change statement
20 from 81 to x2. I got the picture on the right
with N=133. So the area under
the curve y = x2 from
0 to 1 was 133/400, very, very close to 1/3 = 133/399, which is the integral of x2,
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
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.
Finding the area under curves on graph paper: (From the scan
of Sean's graph below I
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*12 or
From 0 to 2 he correctly predicted 1/3 of the 2x22 rectangle
or 1/3*23 (the green rectangle
above) as the area under the curve y = x2 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
A0-1 (x2) = 1/3*13.
Then the area from 0-2 he wrote as A0-2 (x2)
= 1/3*23. 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 0-3? 0-4? 0-n?
How about finding the area under the curve y = x3 from
0-1? 0-2? 0-n?... Look for patterns!