Length of Curve Integral Calculator

Calculate the arc length of a curve using integral calculus. This tool computes the exact length of a curve defined by a function between two points using the arc length formula.

How to Use This Calculator

To calculate the length of a curve:

Enter the function of the curve (e.g., y = x²)
Specify the lower and upper bounds (a and b)
Click "Calculate" to compute the arc length
View the result and visualization

The calculator uses numerical integration to approximate the arc length when an exact solution isn't possible.

The Formula

The length L of a curve y = f(x) from x = a to x = b is given by:

L = ∫_a^b √(1 + (dy/dx)²) dx

Where dy/dx is the derivative of y with respect to x. For parametric curves (x = g(t), y = h(t)), the formula becomes:

L = ∫_a^b √((dx/dt)² + (dy/dt)²) dt

Note: For complex functions, the calculator uses numerical integration with a small step size for accuracy.

Worked Example

Let's find the length of the curve y = x² from x = 0 to x = 1.

First, find the derivative: dy/dx = 2x
Plug into the formula: L = ∫₀¹ √(1 + (2x)²) dx
This integral doesn't have an elementary antiderivative, so we use numerical methods
The calculator computes this as approximately 1.1547 units

Interpreting Results

The result shows the exact arc length when possible, or a numerical approximation when exact calculation isn't feasible. The visualization helps understand the curve's shape relative to its length.

For practical applications, consider:

Rounding to a reasonable number of decimal places
Comparing with other curve lengths
Verifying with known results for simple curves

FAQ

What if the function isn't differentiable?

The calculator will use numerical methods to approximate the derivative where needed. For piecewise functions, ensure the function is continuous at the boundaries.

How accurate are the results?

For exact solutions, results are precise. For numerical approximations, accuracy depends on the step size used in integration (typically 0.001).

Can I use this for 3D curves?

This calculator is for 2D curves only. For 3D space curves, you would need a different formula involving three variables.