Calculated Result
| Function | — |
|---|---|
| Derivative | — |
| Interval | — |
| Integrand | — |
Calculator Input
Enter a function and its derivative. The tool evaluates arc length numerically over your chosen interval and plots the curve with sampled points.
Use JavaScript-style expressions such as sin(x), cos(x), exp(x), log(x), and sqrt(x).
Curve and Integrand Plot
Example Data Table
This sample shows typical inputs and approximate outputs for quick reference. Your exact result can change with interval size and numerical method.
| Function | Derivative | Interval | Method | Approx. Arc Length |
|---|---|---|---|---|
| y = x² | 2x | [0, 2] | Simpson | 4.646784 |
| y = sin(x) | cos(x) | [0, π] | Simpson | 3.820198 |
| y = eˣ | eˣ | [0, 1] | Trapezoidal | 2.003497 |
| y = ln(x) | 1/x | [1, 3] | Midpoint | 2.388401 |
Formula Used
For a curve written as y = f(x), the arc length from x = a to x = b is:
L = ∫[a to b] √(1 + (f'(x))²) dx
The calculator evaluates the integrand √(1 + (f'(x))²) numerically using Simpson's rule, the trapezoidal rule, or the midpoint rule.
When Simpson's rule is selected, the interval is split into an even number of subintervals and weighted as:
L ≈ (h/3) × [g(x₀) + 4g(x₁) + 2g(x₂) + ... + 4g(xₙ₋₁) + g(xₙ)]
Here, g(x) = √(1 + (f'(x))²) and h = (b - a)/n.
How to Use This Calculator
- Enter the function f(x) for plotting.
- Enter the derivative f'(x) exactly as an expression in x.
- Set the lower and upper bounds of integration.
- Choose a numerical method and the number of subintervals.
- Pick the decimal precision and plot sample count.
- Click Calculate Arc Length to show the result above the form.
- Review the graph, result table, and estimated error.
- Use the CSV or PDF buttons to export the summary.
Frequently Asked Questions
1) What does this calculator compute?
It estimates the length of a curve over a chosen interval. You provide the function for plotting and its derivative for the arc length integral.
2) Why do I need to enter the derivative?
The arc length formula uses f'(x) directly. This version expects the derivative because it improves control, speed, and transparency for advanced users.
3) Which numerical method is best?
Simpson's rule is usually the most accurate for smooth curves. Trapezoidal and midpoint methods are useful for comparison and quick checks.
4) Why should subintervals be even for Simpson's rule?
Simpson's rule groups intervals in pairs. An even number of subintervals keeps the weighting pattern valid and preserves the method's standard accuracy.
5) What causes invalid results?
Invalid expressions, undefined derivatives, or intervals outside a function's domain can break evaluation. Logarithms, square roots, and divisions need valid inputs.
6) Is the result exact?
No. The result is a numerical approximation. Accuracy improves when the derivative is correct and the interval is sampled with enough subintervals.
7) What does estimated error mean?
The tool compares two approximations with different step sizes. That difference is a practical estimate, not a strict theoretical error bound.
8) Can I export my result?
Yes. Use the CSV button for spreadsheet-friendly output or the PDF button for a simple printable summary of the current calculation.
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