How Do You Calculate Simpson's Rule?

Simpson's Rule is a numerical method used to approximate the definite integral of a function. It offers a more accurate estimation compared to methods like the Trapezoidal Rule by fitting parabolic arcs to segments of the function's curve.

Understanding Simpson's Rule

Simpson's Rule, also known as Simpson's 1/3 Rule, approximates the area under a curve by dividing the integration region into an even number of subintervals and fitting a parabola through sets of three consecutive points. This approach typically yields a more precise approximation for definite integrals, especially for functions that can be well-approximated by parabolas over short intervals.

Steps to Calculate Simpson's Rule

To calculate Simpson's Rule, you first need to define the integral's parameters and then apply a specific formula. The process involves identifying the number of divisions (n), calculating the width of each subinterval (Δx), and determining the specific input values that the formula will use based on Δx and the integral's upper and lower limits.

Step 1: Define the Integral and Determine Parameters

Begin by identifying the components of the definite integral you wish to approximate:

The function: f(x)
The lower limit of integration: a
The upper limit of integration: b
The number of subintervals: n. It is crucial that n is an even integer (e.g., 2, 4, 6, ...). This is because each parabolic segment used in the approximation spans two subintervals.

Step 2: Calculate Delta x (Width of Subintervals)

Delta x (or Δx) represents the width of each subinterval. Calculate it using the formula:

Δx = (b - a) / n

Step 3: Determine the Input Values (x-coordinates)

Identify the x-coordinates at which you will evaluate the function. These points divide the integration region into n equal subintervals.

x_0 = a
x_1 = a + Δx
x_2 = a + 2Δx
...
x_i = a + iΔx
...
x_n = b

Step 4: Evaluate the Function at Each Input Value

Calculate the value of the function f(x) at each of the x_i coordinates determined in Step 3.

f(x_0)
f(x_1)
f(x_2)
...
f(x_n)

Step 5: Apply Simpson's Rule Formula

Finally, substitute the calculated values into the Simpson's Rule formula:

∫[a,b] f(x) dx ≈ (Δx / 3) * [f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + ... + 2f(x_{n-2}) + 4f(x_{n-1}) + f(x_n)]

Key Pattern of Coefficients:
Notice the pattern of coefficients within the brackets:

The first and last terms (f(x_0) and f(x_n)) have a coefficient of 1.
Terms with odd indices (f(x_1), f(x_3), etc.) have a coefficient of 4.
Terms with even indices (f(x_2), f(x_4), etc., excluding f(x_0) and f(x_n)) have a coefficient of 2.

Practical Insights and Tips

Why n Must Be Even: Simpson's Rule relies on fitting parabolas, each of which requires three points. To cover the entire interval with non-overlapping parabolic segments, you need an even number of subintervals. Each pair of subintervals forms one segment for a parabola.
Accuracy vs. n: Increasing the number of subintervals (n) generally leads to a more accurate approximation of the definite integral. However, it also increases the computational effort.
Comparison to Trapezoidal Rule: Simpson's Rule is often preferred over the Trapezoidal Rule because it provides a more accurate approximation for a given n due to its use of parabolic rather than linear approximations.

Example Calculation: Approximating an Integral

Let's approximate the definite integral of f(x) = x^2 from a=0 to b=2 using Simpson's Rule with n=4.

Problem: ∫[0,2] x^2 dx

Identify Parameters:
- f(x) = x^2
- a = 0
- b = 2
- n = 4 (an even number, so valid)
Calculate Delta x:
- Δx = (b - a) / n = (2 - 0) / 4 = 2 / 4 = 0.5
Determine x-coordinates:
- x_0 = 0
- x_1 = 0 + 0.5 = 0.5
- x_2 = 0 + 2(0.5) = 1.0
- x_3 = 0 + 3(0.5) = 1.5
- x_4 = 0 + 4(0.5) = 2.0
Evaluate f(x) at each x-coordinate:

i x_i f(x_i) = x_i^2

0 0 0^2 = 0

1 0.5 0.5^2 = 0.25

2 1.0 1.0^2 = 1

3 1.5 1.5^2 = 2.25

4 2.0 2.0^2 = 4
Apply Simpson's Rule Formula:

∫[0,2] x^2 dx ≈ (Δx / 3) * [f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + f(x_4)]
∫[0,2] x^2 dx ≈ (0.5 / 3) * [0 + 4(0.25) + 2(1) + 4(2.25) + 4]
∫[0,2] x^2 dx ≈ (1/6) * [0 + 1 + 2 + 9 + 4]
∫[0,2] x^2 dx ≈ (1/6) * [16]
∫[0,2] x^2 dx ≈ 16 / 6 = 8 / 3

`i`	`x_i`	`f(x_i) = x_i^2`
0	0	`0^2 = 0`
1	0.5	`0.5^2 = 0.25`
2	1.0	`1.0^2 = 1`
3	1.5	`1.5^2 = 2.25`
4	2.0	`2.0^2 = 4`

The exact value of ∫[0,2] x^2 dx is [x^3 / 3] from 0 to 2 = (2^3 / 3) - (0^3 / 3) = 8/3. In this specific case, Simpson's Rule with n=4 provides the exact answer because the function f(x) = x^2 is a parabola, and Simpson's Rule is exact for polynomials up to degree 3.