Simpson Rule Solver
Enter a function in terms of x, choose a Simpson method, and review the weighted table, estimate, and error metrics.
Example Data Table
This worked example uses f(x)=x^2 on the interval [0, 4] with 4 subintervals under Composite Simpson 1/3.
| Input | Example Value | Meaning |
|---|---|---|
| Function | x^2 | Quadratic curve to integrate numerically |
| Lower Bound | 0 | Starting point of the interval |
| Upper Bound | 4 | Ending point of the interval |
| Subintervals | 4 | Even count required for the 1/3 rule |
| Step Size h | 1 | Computed as (b - a) / n |
| Simpson Estimate | 21.333333 | Matches the exact integral for this polynomial |
| Exact Value | 21.333333 | Used to confirm zero approximation error |
Formula Used
Composite Simpson 1/3 Rule
For an even number of subintervals, h = (b - a) / n and
∫ab f(x) dx ≈ (h / 3)[f(x0) + 4f(x1) + 2f(x2) + ... + 4f(xn-1) + f(xn)].
Composite Simpson 3/8 Rule
For subinterval counts that are multiples of three, h = (b - a) / n and
∫ab f(x) dx ≈ (3h / 8)[f(x0) + 3f(x1) + 3f(x2) + 2f(x3) + ... + f(xn)].
The calculator also reports the weighted sum, absolute error, relative error, and average value over the interval. If you do not enter an exact value, it builds a dense reference estimate for comparison.
How to Use This Calculator
- Enter your function using
x. Standard functions likesin(x),sqrt(x), andexp(x)are supported. - Provide the lower bound
aand upper boundbfor the definite integral. - Choose Composite Simpson 1/3 or Composite Simpson 3/8, then enter a valid subinterval count.
- Pick the decimal precision that you want in the output.
- Optionally enter an exact value if you already know the true integral and want direct error reporting.
- Press the calculate button. The result appears above the form, followed by a detailed weighted table and export buttons.
FAQs
1. What does Simpson Rule estimate?
It estimates a definite integral by replacing the curve with short quadratic or cubic segments across evenly spaced sample points.
2. When should I use Simpson 1/3?
Use Simpson 1/3 when your subinterval count is even. It is widely used for smooth functions and often gives strong accuracy.
3. When is Simpson 3/8 better?
Simpson 3/8 is useful when the interval division naturally fits multiples of three or when you want an alternative higher-order panel pattern.
4. Why must the points be evenly spaced?
These formulas are derived from equally spaced interpolation points. Unequal spacing breaks the standard coefficients and changes the method entirely.
5. What if I leave the exact value blank?
The solver creates a dense internal reference estimate. That gives a practical error check, though it is not the same as a symbolic exact integral.
6. Can this handle trigonometric and exponential functions?
Yes. Expressions such as sin(x), cos(x), exp(x), sqrt(x), abs(x), and many common combinations are supported.
7. Why does the solver adjust my subinterval count?
If automatic fixing is enabled, the calculator moves invalid values to the next allowed count so the selected Simpson method remains valid.
8. What is included in the exported report?
The export includes the chosen function, interval, method, step size, estimates, error metrics, and the detailed weighted term table.