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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

Assessment:

• Collect activities from each group

• Monitor student progress during completion of activity

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.

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:

[pic]

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’ (points on or under the curve) and use the ratio [pic] to estimate the area.

This is known as the Monte Carlo method for estimating the area under a curve.

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 [pic]and above the x-axis on the interval [pic].

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 [pic]. To find the value, we evaluate [pic]when [pic].

[pic]

[pic] Thus, the maximum y-value for the region is 8.

Step 2: Set the window on the calculator, enter the equation of the function, and graph.

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.

Step 4: Generate 30 random numbers between 0 and 8 for the y-coordinates and store them in [pic]. 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)

Step 6: Count the number of hits generated by your random points. (A ‘hit’ is any point on or below the curve). The ratio [pic] * the area of the viewing rectangle will be the estimated area of the region. (Why is the denominator 30? ________________________)

In the example above, there are 15 hits, therefore, the estimated area is:

[pic] = 16.

(Where did the 4*8 come from? ________________________________ (Hint: [pic]))

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 |Trial 3 |Average Area for 3 |Group’s Average Area |

| | | | |Trials. | |

|Hits | | | | | |

|Attempts |30 |30 |30 | | |

|Estimated | | | | | |

|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 [pic]and [pic]. 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 [pic] and above the x-axis on the interval [pic].

XMIN __________ YMIN __________

XMAX __________ YMAX __________

Expression for random values of x _____________________________________

Expression for random values of y _____________________________________

| |Trial 1 |Trial 2 |Trial 3 |Average Area for 3 |Group’s Average Area |

| | | | |Trials. | |

|Hits | | | | | |

|Attempts |30 |30 |30 | | |

|Estimated | | | | | |

|Area | | | | | |

Notice the shape of this region is a trapezoid (it may help to rotate the calculator [pic]). 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?

________________________________________________________________________

2. Estimate the area under the function [pic] and above the x-axis on the interval [pic].

XMIN __________ YMIN __________

XMAX __________ YMAX __________

Expression for random values of x _____________________________________

Expression for random values of y _____________________________________

| |Trial 1 |Trial 2 |Trial 3 |Average Area for 3 |Group’s Average Area |

| | | | |Trials. | |

|Hits | | | | | |

|Attempts |30 |30 |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?

________________________________________________________________________

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 [pic] and above the x-axis on the interval [pic].

XMIN __________ YMIN __________

XMAX __________ YMAX __________

Expression for random values of x _____________________________________

Expression for random values of y _____________________________________

| |Trial 1 |Trial 2 |Trial 3 |Average Area for 3 |Group’s Average Area |

| | | | |Trials. | |

|Hits | | | | | |

|Attempts |30 |30 |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?

________________________________________________________________________

4. Estimate the area under the function [pic] and above the x-axis on the interval [pic].

XMIN __________ YMIN __________

XMAX __________ YMAX __________

Expression for random values of x _____________________________________

Expression for random values of y _____________________________________

| |Trial 1 |Trial 2 |Trial 3 |Average Area for 3 |Group’s Average Area |

| | | | |Trials. | |

|Hits | | | | | |

|Attempts |30 |30 |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 [pic] and above the x-axis on the interval [pic].

XMIN __________ YMIN __________

XMAX __________ YMAX __________

Expression for random values of x _____________________________________

Expression for random values of y _____________________________________

| |Trial 1 |Trial 2 |Trial 3 |Average Area for 3 |Group’s Average Area |

| | | | |Trials. | |

|Hits | | | | | |

|Attempts |30 |30 |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?

________________________________________________________________________

________________________________________________________________________

Additional Examples and Extension:

6. Estimate the area under the function [pic] and above the x-axis on the interval [pic].

XMIN __________ YMIN __________

XMAX __________ YMAX __________

Expression for random values of x _____________________________________

Expression for random values of y _____________________________________

| |Trial 1 |Trial 2 |Trial 3 |Average Area for 3 |Group’s Average Area |

| | | | |Trials. | |

|Hits | | | | | |

|Attempts |30 |30 |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 [pic] and above the x-axis on the interval [pic].

XMIN __________ YMIN __________

XMAX __________ YMAX __________

Expression for random values of x _____________________________________

Expression for random values of y _____________________________________

| |Trial 1 |Trial 2 |Trial 3 |Average Area for 3 |Group’s Average Area |

| | | | |Trials. | |

|Hits | | | | | |

|Attempts |30 |30 |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?

________________________________________________________________________

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