Understanding Simpson’s Rule in Excel – With Practical Example

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In the world of data analysis, engineering, and applied mathematics, integration is often required to calculate areas under curves. When dealing with complex functions or raw tabulated data, traditional calculus may not be feasible — and that’s where Simpson’s Rule comes in.

Excel provides a great platform to apply this technique using formulas, even without using advanced programming.

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🔍 What is Simpson’s Rule?

Simpson’s Rule is a numerical method that approximates the definite integral of a function by estimating the area under the curve using parabolic arcs rather than straight lines (as in the trapezoidal rule). It provides higher accuracy, especially when the data or function changes curvature.

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✅ Simpson’s Rule Formula

For a function f(x)f(x) defined on interval [a,b][a, b], divided into n even sub-intervals, Simpson’s Rule is: ∫abf(x)dx≈h3[f(x0)+4f(x1)+2f(x2)+4f(x3)+⋯+4f(xn−1)+f(xn)]\int_a^b f(x)dx \approx \frac{h}{3} \left[ f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + \dots + 4f(x_{n-1}) + f(x_n) \right]

Where:

• h=b−anh = \frac{b – a}{n}
• nn is even
• x0,x1,…,xnx_0, x_1, …, x_n are equally spaced data points

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💼 Real-World Example in Excel

Let’s apply Simpson’s Rule to estimate the following integral: ∫0411+x2dx\int_0^4 \frac{1}{1 + x^2} dx

This is the integral of the arctangent function, which cannot be easily integrated manually.

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🧮 Step-by-Step in Excel

1. Create the x values (A2:A6)
   You divide the interval [0, 4] into 4 equal parts (n = 4):
   0, 1, 2, 3, 4
2. Create the corresponding f(x) values in column B
   Formula: =1 / (1 + A2^2), then fill down.

A (x)	B = f(x) = 1/(1+x²)
0	1.0000
1	0.5000
2	0.2000
3	0.1000
4	0.0588

3. Calculate h
   Formula: = (A6 - A2) / 4 Result: 1
4. Apply Simpson’s Rule in Excel = (1/3) * (B2 + 4*B3 + 2*B4 + 4*B5 + B6) Result: Approx. 1.3255

This value is very close to the actual integral of arctangent(4) ≈ 1.3258, showcasing Simpson’s accuracy.

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✨ Where Can You Use Simpson’s Rule in Excel?

• When working with experimental data from labs or sensors
• To approximate areas under curves in physics, finance, biology, and statistics
• When analytical integration is too complex or not possible
• For students and professionals who need quick, reliable estimations

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