Area between curves
This is a parabolic function:
What is its graph?
If you get infinite points you will have the graph.
We don't have infinite time to draw infinite points. However, we have Geogebra!!
How we can get the area between the graph and the x-axis in a closed interval?
How do you think that we can measure the area? Write your answer.
We can use rice grains to measure the area. Look at the next photo!
How do you can measure a mathematical area with rice grains?
- Print the graphic on paper and spread glue over the rectangle (Rectangle Area: 8 units square).
- Throw alleatory rice grains over the rectangle.
- Repeat the process until you have finished with all the grains.
- Wait a while for the glue to dry.
- Count the number of grains over the region between the function and the x-axis (success number: F). Add to F the number of grains situated over the rest of the rectangle (total number: N).
- If N is a big number, we can assume that the relative frecuency F/N tend to the probability (P) of finding a rice grain on the area created by the function and the x-axis, in the closed interval [0,2].
- Multiply to get the solution area.
- What is your result?
Geogebra can create bunch of rice grains to measure the area!!
If Start = 0, Finish = 2 and maxPoints = 10000, what value does the probability tend to? Write your answer.
Challenge: Design a Geogebra simulation with a parabolic function, a closed interval and 500 points to estimate the area
Hints for the task!
Function:
Interval:
[0,2]
Image of the interval:
[0,4]
Slider numPoints:
from 1 to 500
Random horizontal coordinate of a point from x=0,001 to x=2:
aleaHor = AleatorioEntre(1,2000)/1000
Random vertical coordinate of a point from y=0,001 to y=4:
aleaVer = AleatorioEntre(1,4000)/1000
Create a list of random horizontal coordinates:
coordHor = Secuencia(aleaHor, i , 1, 500)
Create a list of random vertical coordinates:
coordVer = Secuencia(aleaVer, i , 1, 500)
Create a list of points:
points = Secuencia((coordHor(i), coordVer(i)),i,1,500)
Watch out! Compare the vertical coordinate of each point with the image of the function generated by the horizontal coordinate:
count = Secuencia(Si(y(points(i)) < f(x(points(i))), true, false), i, 1, numPoints)
How do we know the number of points that belong to the solution area?
Commands: Secuencia() and Si(condition, then, else)
We need to compare the vertical coordinate of each point with the image of each horizontal coordinate in the function.
Probability:
prob = Suma(count)/500
Solution area:
areaSolution = 8·prob