Finding the area under a curve with random probability.
(The Monte Carlo method for estimating area under curve)
Summary: Initially, students will graph a curve whose area can be found using
Geometry methods using the Monte Carlo method that uses random points
and probability to estimate the area under the curve. Students also calculate
the area geometrically to prove that the method provides a reasonable
estimate of area. Students then use the method to estimate the area under
non-geometric curves whose exact area is found using calculus methods.
Key Words:
Monte Carlo, Area Under Curve, Probability Ratios
Background Knowledge:
 Geometry area formulas
 Key strokes for random number, list, store, window setting
 Ratios and percentages
 Calculation of mean
OACS Standards:
Mathematical Process Measurement
 Standard G: Write clearly and coherently about mathematical thinking and ideas.
Measurement Standard
 Standard C - Grades 8-10: Apply indirect measurement techniques, tools and
formulas, as appropriate to find perimeter, circumference and area of circles,
triangles, quadrilaterals and composite shapes and to find volume of prisms,
cylinders and pyramids
 Standard C - Grades 11-12: Estimate and compute areas and volume in
increasingly complex problem situations
Learning Objectives:
1. The student will find the area under a curve using geometry formulas
2. Students will apply the Monte Carlo method to estimate the area under a curve on
a given interval
3. Students will make comparisons between the estimated area and the actual area
Materials:
 Graphing calculator
 Copy of inquiry based activity
Suggested Procedures:
 Use soft dart board and set it on wall to simulate probability of hitting area of
target (“Attention Getter”)
 Group students in threes so students can compare individual results and create a
group average for increased accuracy
Project AMP
Dr. Antonio R. Quesada Director, Project AMP
Assessment:
 Collect activities from each group
 Monitor student progress during completion of activity
Project AMP
Dr. Antonio R. Quesada Director, Project AMP
Name: ___________________
ACTIVITY: MONTE CARLO METHOD
FOR ESTIMATING THE AREA UNDER THE CURVE
Lesson Objectives:
 Reinforce concept of domain and codomain
 Calculate area under a curve on a given interval
 Determine percent of error
 Introduce integration for calculus
In geometry, we have used the ratio of areas to calculate the probability that a dart would
hit a shaded region within a given target. For example, suppose you want to win a
stuffed animal for your mom’s birthday and decide to take a chance on the dart game.
The game consists of 20 balloons (all congruent of course!) on a rectangular board as in
Figure 1. If you throw a dart and it hits a balloon you win a prize.
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The probability of randomly hitting a balloon is the ratio
of the area of all the balloons to the area of the entire
rectangular region:
Area of balloons
P(hit balloon) =
Area of board
Figure 1
To simulate this activity, we could throw darts at a ‘board’ and
count the number of hits to determine the probability and then
multiply the area of the board by this probability to determine the
area of the target region, in this case the balloons. We are going to
use this process to estimate the area of regions whose exact area
cannot be calculated without the use of calculus.
For each problem you are given an equation of a curve and an x-interval that together
begin to define our region (board). You need to determine the y-interval that captures the
region and set an appropriate viewing window based on these x and y-intervals on the
calculator.
Once the region, is set create 30 random ordered pairs with this region and create a
scatter plot to represent where the ‘darts’ hit the board. Next, count the number of ‘hits’
hits
(points on or under the curve) and use the ratio
to estimate the area.
attempts
This is known as the Monte Carlo method for estimating the area under a curve.
Project AMP
Dr. Antonio R. Quesada Director, Project AMP
The first two problems are examples that can easily be solved geometrically in order to
compare estimated area to actual area and verify the method used.
Example:
Estimate the area under the function y  2 x  8 and above the x-axis on the interval
0  x  4.
Solution:
Step 1: Determine the appropriate y-interval. Since we want the area above x-axis, the
minimum y-value is 0. Also, since the graph is decreasing (How do know the function is
decreasing? ________________________________________________), the maximum
y-value will be at the left of the region along the boundary where x  0 . To find the
value, we evaluate y  2 x  8 when x  0 .
y  2(0)  8
y  08
Thus, the maximum y-value for the region is 8.
y 8
Step 2: Set the window on the calculator, enter the equation of the function, and graph.
Figure 2
Figure 3
Figure 4
Looking at the window, the x-values of the points representing the ‘darts’ need to be
between 0 and 4 and the y-values need to be between 0 and 8.
Step 3: The 30 coordinate points in our region are generated randomly. To generate 30
x-values, enter the key sequence [4] [MATH] PRB [1] [(] [3] [0] [)] [STO][L1]. This
statement will generate 30 random numbers with values between 0 and 4 and store them
in L1 as x-values.
Project AMP
Dr. Antonio R. Quesada Director, Project AMP
Step 4: Generate 30 random numbers between 0 and 8 for the y-coordinates and store
them in L 2 . The screen should look like Figure 5 with the exception that the numbers in
your list will be different because the numbers are randomly created by the calculator.
Step 5: Turn ON the STAT PLOT and identify L1 and L2. (Figure 6) Graph these points
with your equations. (Figure 7)
Figure 6
Figure 5
Figure 7
Step 6: Count the number of hits generated by your random points. (A ‘hit’ is any point
hits
on or below the curve). The ratio
* the area of the viewing rectangle will be the
30
estimated area of the region. (Why is the denominator 30? ________________________)
In the example above, there are 15 hits, therefore, the estimated area is:
Area under curve =
15
*(4*8) = 16.
30
(Where did the 4*8 come from? ________________________________ (Hint: A=lw ))
Step 7: Record your results in the chart below. Repeat the experiment two times by
reentering your two random number generator expressions for the x-values and y-values
and complete the table. Calculate the average of your three trials then find the average of
your group’s trials and enter in the appropriate boxes below.
Trial 1
Trial 2
30
30
Trial 3
Hits
Attempts
Average
Area for 3
Trials.
30
Estimated
Area
Project AMP
Dr. Antonio R. Quesada Director, Project AMP
Group’s
Average
Area
Step 8: For this example, the exact area under the curve can be calculated geometrically
to test the results of your estimated area. Notice the shape of the target region is a
triangle with b  4 and h  8 . Calculate the exact area.
Exact area under the curve is: _______________
Debrief Questions:
How does your average area estimate compare to actual? What is the percent of error
between your estimated area and the actual area?
________________________________________________________________________
How does your average area estimate compare to the group’s average? What is the
percent of error between your group’s estimated area and the actual area?
________________________________________________________________________
What accounts for the differences between your average and the group’s average?
________________________________________________________________________
Additional Examples:
1. Estimate the area under the function y  2 x  3 and above the x-axis on the
interval 0  x  3 .
XMIN __________
YMIN __________
XMAX __________
YMAX __________
Expression for random values of x _____________________________________
Expression for random values of y _____________________________________
Trial 1
Trial 2
30
30
Trial 3
Hits
Attempts
Average
Area for 3
Trials.
Group’s
Average
Area
30
Estimated
Area
Notice the shape of this region is a trapezoid (it may help to rotate the calculator 90 ).
Calculate the area of the region geometrically and compare to your estimate.
Exact area under the curve is: _______________
Project AMP
Dr. Antonio R. Quesada Director, Project AMP
Debrief Questions:
How does your average area estimate compare to actual? What is the percent of error
between your estimated area and the actual area?
________________________________________________________________________
How does your average area estimate compare to the group’s average? What is the
percent of error between your group’s estimated area and the actual area?
________________________________________________________________________
What accounts for the differences between your average and the group’s average?
________________________________________________________________________
2. Estimate the area under the function y  2 x 2  1 and above the x-axis on the
interval 0  x  2 .
XMIN __________
YMIN __________
XMAX __________
YMAX __________
Expression for random values of x _____________________________________
Expression for random values of y _____________________________________
Trial 1
Trial 2
30
30
Trial 3
Hits
Attempts
Average
Area for 3
Trials.
Group’s
Average
Area
30
Estimated
Area
The area of this region cannot be determined geometrically. When you have completed
the worksheet check with your teacher for the actual area.
Exact area under the curve is: _______________
Debrief Questions:
How does your average area estimate compare to actual? What is the percent of error
between your estimated area and the actual area?
________________________________________________________________________
Project AMP
Dr. Antonio R. Quesada Director, Project AMP
How does your average area estimate compare to the group’s average? What is the
percent of error between your group’s estimated area and the actual area?
________________________________________________________________________
What accounts for the differences between your average and the group’s average?
________________________________________________________________________
3. Estimate the area under the function y  x and above the x-axis on the
interval 0  x  4 .
XMIN __________
YMIN __________
XMAX __________
YMAX __________
Expression for random values of x _____________________________________
Expression for random values of y _____________________________________
Trial 1
Trial 2
30
30
Trial 3
Hits
Attempts
Average
Area for 3
Trials.
Group’s
Average
Area
30
Estimated
Area
In this case, the area of this region cannot be determined geometrically. Check with your
teacher for the actual area.
Exact area under the curve is: _______________
Debrief Questions:
How does your average area estimate compare to actual? What is the percent of error
between your estimated area and the actual area?
________________________________________________________________________
How does your average area estimate compare to the group’s average? What is the
percent of error between your group’s estimated area and the actual area?
________________________________________________________________________
What accounts for the differences between your average and the group’s average?
________________________________________________________________________
Project AMP
Dr. Antonio R. Quesada Director, Project AMP
4. Estimate the area under the function y  x 3  2 and above the x-axis on the
interval 0  x  2 .
XMIN __________
YMIN __________
XMAX __________
YMAX __________
Expression for random values of x _____________________________________
Expression for random values of y _____________________________________
Trial 1
Trial 2
30
30
Trial 3
Hits
Attempts
Average
Area for 3
Trials.
Group’s
Average
Area
30
Estimated
Area
In this case, the area of this region cannot be determined geometrically. Check with your
teacher for the actual area.
Exact area under the curve is: _______________
Debrief Questions:
How does your average area estimate compare to actual? What is the percent of error
between your estimated area and the actual area?
________________________________________________________________________
How does your average area estimate compare to the group’s average? What is the
percent of error between your group’s estimated area and the actual area?
________________________________________________________________________
What accounts for the differences between your average and the group’s average?
________________________________________________________________________
5. Estimate the area under the function y   x 2  25 and above the x-axis on
the interval 0  x  5 .
XMIN __________
YMIN __________
XMAX __________
YMAX __________
Expression for random values of x _____________________________________
Expression for random values of y _____________________________________
Project AMP
Dr. Antonio R. Quesada Director, Project AMP
Trial 1
Trial 2
30
30
Trial 3
Hits
Attempts
Average
Area for 3
Trials.
Group’s
Average
Area
30
Estimated
Area
In this case, the area of this region cannot be determined geometrically. Check with your
teacher for the actual area.
Exact area under the curve is: _______________
Debrief Questions:
How does your average area estimate compare to actual? What is the percent of error
between your estimated area and the actual area?
________________________________________________________________________
How does your average area estimate compare to the group’s average? What is the
percent of error between your group’s estimated area and the actual area?
________________________________________________________________________
What accounts for the differences between your average and the group’s average?
________________________________________________________________________
Summary Questions:
What was the main goal of this activity?
________________________________________________________________________
________________________________________________________________________
How does this connect with topics from previous courses?
________________________________________________________________________
________________________________________________________________________
Project AMP
Dr. Antonio R. Quesada Director, Project AMP
Additional Examples and Extension:
6. Estimate the area under the function y  x and above the x-axis on the
interval 5  x  5 .
XMIN __________
YMIN __________
XMAX __________
YMAX __________
Expression for random values of x _____________________________________
Expression for random values of y _____________________________________
Trial 1
Trial 2
30
30
Trial 3
Hits
Attempts
Average
Area for 3
Trials.
Group’s
Average
Area
30
Estimated
Area
Calculate the area of the region geometrically and compare to your estimate.
Exact area under the curve is: _______________
Debrief Questions:
How does your average area estimate compare to actual? What is the percent of error
between your estimated area and the actual area?
________________________________________________________________________
How does your average area estimate compare to the group’s average? What is the
percent of error between your group’s estimated area and the actual area?
________________________________________________________________________
What accounts for the differences between your average and the group’s average?
________________________________________________________________________
7. Estimate the area under the function y  sin( x ) and above the x-axis on the
interval 0  x  120 .
XMIN __________
YMIN __________
XMAX __________
YMAX __________
Expression for random values of x _____________________________________
Expression for random values of y _____________________________________
Project AMP
Dr. Antonio R. Quesada Director, Project AMP
Trial 1
Trial 2
30
30
Trial 3
Hits
Attempts
Average
Area for 3
Trials.
Group’s
Average
Area
30
Estimated
Area
In this case, the area of this region cannot be determined geometrically. Check with your
teacher for the actual area.
Exact area under the curve is: _______________
Debrief Questions:
How does your average area estimate compare to actual? What is the percent of error
between your estimated area and the actual area?
________________________________________________________________________
How does your average area estimate compare to the group’s average? What is the
percent of error between your group’s estimated area and the actual area?
________________________________________________________________________
What accounts for the differences between your average and the group’s average?
________________________________________________________________________
Project AMP
Dr. Antonio R. Quesada Director, Project AMP