Calculator Inputs
Enter explicit multiplication like 2*x and powers like x^2.
Example Data Table
Example case: use y = f(x), enter dy/dx = x/3, lower bound 0, upper bound 6, and Simpson integration.
| x | dy/dx | √(1 + (dy/dx)2) | Comment |
|---|---|---|---|
| 0.00000 | 0.00000 | 1.00000 | Starting point |
| 1.50000 | 0.50000 | 1.11803 | Moderate local stretch |
| 3.00000 | 1.00000 | 1.41421 | Slope equals one |
| 4.50000 | 1.50000 | 1.80278 | Steeper region |
| 6.00000 | 2.00000 | 2.23607 | Largest shown stretch |
Formula Used
For curves written as y = f(x), the arc length from x = a to x = b is:
L = ∫ab √(1 + (dy/dx)2) dx
For curves written as x = g(y), the arc length from y = c to y = d is:
L = ∫cd √(1 + (dx/dy)2) dy
This page evaluates the derivative numerically at many sample points. It then estimates the integral using Simpson or trapezoidal summation. Larger segment counts usually improve accuracy, especially when slopes change quickly.
How to Use This Calculator
- Select whether the curve is written as y = f(x) or x = g(y).
- Enter the derivative expression only, such as x/3, 2*x, or sin(x).
- Type the lower and upper bounds for the integration interval.
- Choose Simpson for stronger accuracy or trapezoidal for a simpler estimate.
- Set segment count and decimal places to control detail.
- Press Calculate Arc Length to display the result above the form.
- Review the table and Plotly graph for local behavior.
- Use the export buttons to save the result table and summary.
Expression Tips
- Use explicit multiplication: 2*x, not 2x.
- Use ^ for powers, such as x^2.
- Allowed functions include sin, cos, tan, sqrt, ln, log, exp, and abs.
- log means base-10 logarithm. ln means natural logarithm.
Frequently Asked Questions
1. What does this calculator actually compute?
It estimates the length of a smooth curve over a chosen interval. You enter the derivative, bounds, and a numerical method. The calculator then evaluates the arc-length integral and shows tables, checks, and a graph.
2. Why do I enter the derivative instead of the original function?
The standard arc-length formula depends directly on the derivative. Entering the derivative avoids symbolic differentiation inside the page and keeps the calculation fast, flexible, and reliable for many classroom-style problems.
3. When should I choose Simpson instead of trapezoidal?
Simpson usually gives better accuracy for smooth curves when enough even segments are used. Trapezoidal is simpler and still useful. Comparing both results gives a quick reasonableness check for your chosen settings.
4. What happens if I reverse the bounds?
The calculator automatically reorders the interval so length stays positive and meaningful. A notice appears when this happens. That keeps the output focused on geometric distance rather than signed integral direction.
5. How many segments should I use?
Start around 200 for ordinary problems. Increase the count when the derivative changes quickly, oscillates, or grows sharply. Watch how the main result and the comparison estimate move closer as segments increase.
6. Can I use trigonometric and logarithmic expressions?
Yes. The calculator accepts common functions including sin, cos, tan, sqrt, ln, log, exp, abs, and hyperbolic forms. Use explicit multiplication and standard parentheses so the expression is interpreted correctly.
7. What does the Plotly graph show?
One trace shows the arc-length integrand, which measures local stretching. Another trace shows cumulative length along the interval. Together, they help you see where the curve contributes most to total distance.
8. What do the CSV and PDF downloads include?
The CSV export contains the computed sample table. The PDF export includes the summary values and the same data rows. These files are useful for assignments, checking work, and saving repeatable calculations.