Parameter Setup
Calculate Step-by-Step
Enter a mathematical function (even subintervals) or coordinate table on the left, then click Calculate to view coordinates, quadratic graphs, and errors.
Mathematical Detail: Pi Estimation
Since the definite integral of $\int_0^1 \frac{dx}{1+x^2} = \arctan(1) - \arctan(0) = \frac{\pi}{4}$, multiplying this rule's approximate integration result by $4$ provides an estimation for $\pi$:
Estimated $\pi$: 3.14159265... | True value: $3.1415926535...$
Problem Statement
Simpson's 1/3 Rule Formula
Definite integration is approximated by fitting quadratic parabolas through groups of three adjacent points:
Step-by-Step Derivation
Calculate Interval Step Size ($h$)
The width of each subinterval is determined by splitting the range $[a, b]$ into $n$ equal portions:
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, 4 for odd indices, 2 for even indices):
| $i$ | $x_i = a + ih$ | $y_i = f(x_i)$ | Type |
|---|
Substitute Into Formula
Group indices to find boundary, odd, and even coordinate sums:
Final Approximation
Evaluate the expression to find the final integral value:
Geometric Visualization
The shaded area represents the quadratic parabolic segments integrated under the curve. Move the cursor over points to check numerical coordinates.