Simpson's 3/8 Rule Calculator

Compute numerical integration using cubic curve segments with step-by-step evaluations.

Parameter Setup
Use standard operators (*, /, +, -, ^). Use exp(x) for $e^x$.
Note: Simpson's 3/8 Rule requires $n$ to be a MULTIPLE of 3.
$i$ $x_i$ $y_i$
Ensure $x$ coordinates are equally spaced and sorted, and total intervals $n$ is a multiple of 3 (total rows - 1 is divisible by 3).

Calculate Step-by-Step

Enter a mathematical function (subintervals multiple of 3) or coordinate table on the left, then click Calculate to view coordinates, cubic segment graphs, and errors.

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Approximate Value
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True / Reference Value
0.000000
Absolute Error

Problem Statement

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Simpson's 3/8 Rule Formula

Definite integration is approximated by fitting cubic polynomials through groups of four adjacent points:

$$I = \frac{3h}{8} \left[ y_0 + y_n + 3\sum_{i \neq 3k} y_i + 2\sum_{i = 3k} y_i \right]$$
Step-by-Step Derivation
1
Calculate Interval Step Size ($h$)

The width of each subinterval is determined by splitting the range $[a, b]$ into $n$ equal portions:

2
Partition Coordinates Table

Divide $[a, b]$ using step size $h$ and evaluate the function $y = f(x)$ at each boundary point. Note the weighting coefficients (1 for boundaries, 2 for multiples of 3 indices, and 3 for all other indices):

$i$ $x_i = a + ih$ $y_i = f(x_i)$ Type
3
Substitute Into Formula

Group indices to find boundary, non-divisible-by-3, and divisible-by-3 coordinate sums:

4
Final Approximation

Evaluate the expression to find the final integral value:

Geometric Visualization

The shaded area represents the cubic polynomial segments integrated under the curve. Move the cursor over points to check numerical coordinates.

Recent Calculations
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