How to Calculate the Approximate Area of Irregular Shapes Using Simpson’s 1/3 Rule

Updated on February 8, 2020
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Ray is a Licensed Civil Engineer and specializes in Structural Engineering. He loves to write anything about education.

Introduction to Area Approximation

Are you having trouble solving areas of complicated and irregular shaped curve figures? If yes, this is the perfect article for you. There are a lot of methods and formulas used in approximating the area of irregularly shaped curves, just as shown in the figure below. Among these are Simpson’s Rule, the Trapezoidal Rule, and Durand’s Rule.

Trapezoidal Rule is an integration rule where you divide the total area of the irregular shaped figure into little trapezoids before evaluating the area under a specific curve. Durand’s Rule is a slightly more complicated but more precise integration rule than the trapezoidal rule. This method of area approximation uses the Newton-Cotes formula, which is an extremely useful and straightforward integration technique. Lastly, Simpson’s Rule gives the most accurate approximation as compared to the other two mentioned formulas. It is also important to note that the greater the value of n in Simpson’s Rule, the greater accuracy of area approximation.

What Is Simpson’s 1/3 Rule?

Simpson’s Rule is named after the English mathematician Thomas Simpson who was from Leicestershire England. But for some reason, the formulas used in this method of area approximation were similar to Johannes Kepler’s formulas used over 100 years prior. That is the reason why many mathematicians call this method the Kepler’s Rule.

Simpson’s Rule is considered as a very diverse numerical integration technique. It is entirely based on the type of interpolation you will use. Simpson’s 1/3 Rule or Composite Simpson’s Rule is based upon a quadratic interpolation while Simpson’s 3/8 Rule is based upon a cubic interpolation. Among all methods of area approximation, Simpson’s 1/3 Rule gives the most accurate area because parabolas are used to approximate each part of the curve, and not rectangles or trapezoids.

Area Approximation Using Simpson's 1/3 Rule
Area Approximation Using Simpson's 1/3 Rule | Source

Simpson's 1/3 Rule states that if y0, y1, y2,...,y3 (n is even) are the lengths of a series of parallel chords of uniform interval d, the area of the figure enclosed above is given approximately by the formula below. Note that if the figure ends with points, then take y0 = yn = 0.

A = (1/3)(d) [(y0 + yn) + 4(y1 + y3 + ... + yn-1) + 2(y2 + y4 + ... + yn-2)]

Problem 1

Calculating the Area of Irregular Shapes Using Simpson's 1/3 Rule
Calculating the Area of Irregular Shapes Using Simpson's 1/3 Rule | Source

Solution

a. Given the value of n = 10 of the irregularly shaped figure, identify the height values from y0 to y10. Create a table and list all the height values from left to right for a more organized solution.

Variable (y)
Height Value
y0
10
y1
11
y2
12
y3
11
y4
6
y5
7
y6
4
y7
8
y8
4
y9
3
y10
0
Calculating the Approximate Area of Irregular Shapes Using Simpson’s 1/3 Rule

b. The given value of the uniform interval is d = 0.75. Substitute the height values (y) in the given Simpson's rule equation. The resulting answer is the approximate area of the given shape above.

A = (1/3)(d) [(y0 + yn) + 4(y1 + y3 + ... + yn-1) + 2(y2 + y4 + ... + yn-2)]

A = (1/3)(3) [(10 + 0) + 4(11 + 11 + 7 + 8 + 3) + 2(12 + 6 + 4 + 4)]

A = 222 square units

c. Find the area of the right triangle formed from the irregular shape. Given a height of 10 units and an angle of 30°, find the length of adjacent sides and compute for the area of the right triangle using Scissors formula or Heron's formula.

Length = 10 / tan (30°)

Length = 17.32 units

Hypotenuse = 10 / sin (30°)

Hypotenuse = 20 units

Semi-Perimeter (s) = (10 + 20 + 17.32) / 2

Semi-Perimeter (s) = 23. 66 units

Area (A) = √s (s - a) (s - b) (s - c)

Area (A) = √23.66 (23.66 - 10) (23.66 - 20) (23.66 - 17.32)

Area (A) = 86.6 square units

d. Subtract the area of the right triangle from the area of the whole irregular figure.

Shaded Area (S) = Total Area - Triangular Area

Shaded Area (S) = 222 - 86.6

Shaded Area (S) = 135.4 square units

Final Answer: The approximate area of the irregular figure above is 135.4 square units.

Problem 2

Calculating the Area of Irregular Shapes Using Simpson's 1/3 Rule
Calculating the Area of Irregular Shapes Using Simpson's 1/3 Rule | Source

Solution

a. Given the value of n = 6 of the irregularly shaped figure, identify the height values from y0 to y6. Create a table and list all the height values from left to right for a more organized solution.

Variable (y)
Height Value
y0
5
y1
3
y2
4
y3
6
y4
4.5
y5
1.5
y6
0
Calculating the Approximate Area of Irregular Shapes Using Simpson’s 1/3 Rule

b. The given value of the uniform interval is d = 1.00. Substitute the height values (y) in the given Simpson's rule equation. The resulting answer is the approximate area of the given shape above.

A = (1/3)(d) [(y0 + yn) + 4(y1 + y3 + ... + yn-1) + 2(y2 + y4 + ... + yn-2)]

A = (1/3)(1.00) [(5 + 0) + 4(3 + 6 + 1.5) + 2(4 + 4.5)]

A = 21.33 square units

Final Answer: The approximate area of the irregular figure above is 21.33 square units.

Problem 3

Calculating the Area of Irregular Shapes Using Simpson's 1/3 Rule
Calculating the Area of Irregular Shapes Using Simpson's 1/3 Rule | Source

Solution

a. Given the value of n = 6 of the irregularly shaped figure, identify the height values from y0 to y6. Create a table and list all the height values from left to right for a more organized solution.

Variable (y)
Upper Value
Lower Value
Height Value (Sum)
y0
0
0
0
y1
3
2
5
y2
1.5
1.75
3.25
y3
1.75
4
5.75
y4
3
2.75
5.75
y5
2.75
3
5.75
y6
0
0
0
Calculating the Approximate Area of Irregular Shapes Using Simpson’s 1/3 Rule

b. The given value of the uniform interval is d = 1.50. Substitute the height values (y) in the given Simpson's rule equation. The resulting answer is the approximate area of the given shape above.

A = (1/3)(d) [(y0 + yn) + 4(y1 + y3 + ... + yn-1) + 2(y2 + y4 + ... + yn-2)]

A = (1/3)(1.50) [(0 + 0) + 4(5 + 5.75 + 5.75) + 2(3.25 + 5.75)]

A = 42 square units

Final Answer: The approximate area of the irregular shape above is 42 square units.

Problem 4

Calculating the Area of Irregular Shapes Using Simpson's 1/3 Rule
Calculating the Area of Irregular Shapes Using Simpson's 1/3 Rule | Source

Solution

a. Given the value of n = 8 of the irregularly shaped figure, identify the height values from y0 to y8. Create a table and list all the height values from left to right for a more organized solution.

Variable (y)
Height Value
y0
10
y1
9
y2
8
y3
7
y4
6
y5
5
y6
4
y7
3
y8
0
Calculating the Approximate Area of Irregular Shapes Using Simpson’s 1/3 Rule

b. The given value of the uniform interval is d = 1.50. Substitute the height values (y) in the given Simpson's rule equation. The resulting answer is the approximate area of the given shape above.

A = (1/3)(d) [(y0 + yn) + 4(y1 + y3 + ... + yn-1) + 2(y2 + y4 + ... + yn-2)]

A = (1/3)(1.50) [(10 + 0) + 4(9 + 7 + 5 +3) + 2(8 + 6 +4)]

A = 71 square units

Final Answer: The approximate area of the irregular shape above is 71 square units.

Problem 5

Calculating the Area of Irregular Shapes Using Simpson's 1/3 Rule
Calculating the Area of Irregular Shapes Using Simpson's 1/3 Rule | Source

Solution

a. Given the equation of the irregular curve, identify the height values from y0 to y8 by substituting each value of x to solve for the corresponding value of y. Create a table and list all the height values from left to right for a more organized solution. Use an interval of 0.5.

Variable (y)
X-Value
Height Value
y0
1.0
1.732050808
y1
1.5
1.870828693
y2
2.0
2.0000000
y3
2.5
2.121320344
y4
3.0
2.236067977
y5
3.5
2.34520788
y6
4.0
2.449489743
Calculating the Approximate Area of Irregular Shapes Using Simpson’s 1/3 Rule

b. Use the uniform interval d = 0.50. Substitute the height values (y) in the given Simpson's rule equation. The resulting answer is the approximate area of the given shape above.

A = (1/3)(d) [(y0 + yn) + 4(y1 + y3 + ... + yn-1) + 2(y2 + y4 + ... + yn-2)]

A = (1/3)(0.50) [(1.73 + 2.45) + 4(1.87 + 2.12 + 2.35) + 2(2 + 2.23)]

A = 6.33 square units

Final Answer: The approximate area of the irregular shape above is 6.33 square units.

Problem 6

Calculating the Area of Irregular Shapes Using Simpson's 1/3 Rule
Calculating the Area of Irregular Shapes Using Simpson's 1/3 Rule | Source

Solution

a. Given the value of n = 8 of the irregularly shaped figure, identify the height values from y0 to y8. Create a table and list all the height values from left to right for a more organized solution.

Variable (y)
Height Value
y0
50
y1
40
y2
30
y3
27
y4
28
y5
38
y6
40
y7
45
y8
48
Calculating the Approximate Area of Irregular Shapes Using Simpson’s 1/3 Rule

b. The given value of the uniform interval is d = 5.50. Substitute the height values (y) in the given Simpson's rule equation. The resulting answer is the approximate area of the given shape above.

A = (1/3)(d) [(y0 + yn) + 4(y1 + y3 + ... + yn-1) + 2(y2 + y4 + ... + yn-2)]

A = (1/3)(5.50) [(50 + 48) + 4(40 + 27 +38 +45) + 2(30 +28 + 40)]

A = 1639 square units

Final Answer: The approximate area of the irregular shape above is 1639 square units.

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    © 2020 Ray

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