How to Determine Arc Length Using Integration

Arc length calculations are essential in calculus for determining the exact length of irregular curves. They’re essential in computer graphics for rendering curves, in astronomy for measuring celestial paths, and in biology for analyzing structures like DNA and protein folding.

The arc length formula, step by step

Here’s a step-by-step breakdown:

1. Formula for Arc Length:
   The arc length \( S \) of a curve defined by a function \( y = f(x) \) from \( x = a \) to \( x = b \) is given by:
   \( S = \int_{a}^{b} \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx \)
   If the function is given in terms of \( y \) (i.e., \( x = g(y) \)), the formula is:
   \( S = \int_{c}^{d} \sqrt{1 + \left( \frac{dx}{dy} \right)^2} \, dy \)
2. Find the Derivative:

• Differentiate the function \( f(x) \) to find \( \frac{dy}{dx} \).

3. Substitute into the Formula:

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• Replace \( \frac{dy}{dx} \) in the arc length formula with the derivative you found.

4. Integrate:

• Evaluate the integral from \( a \) to \( b \). This might require numerical methods if the integral can’t be solved analytically.

5. Interpret the Result:

• The result of the integration gives the length of the curve from \( x = a \) to \( x = b \).

Example:

Worked example: y = x² from 0 to 1

Consider a curve \( y = x^2 \) between \( x = 0 \) and \( x = 1 \).

1. Differentiate \( y = x^2 \) to get \( \frac{dy}{dx} = 2x \).
2. The formula becomes \( S = \int_{0}^{1} \sqrt{1 + (2x)^2} \, dx \).
3. Integrate this expression to find the arc length:

\( \dfrac{\operatorname{arsinh}\left(2x\right)}{4}+\dfrac{x\sqrt{4x^2+1}}{2} \)

\( \dfrac{\operatorname{arsinh}\left(2\right)+2\sqrt{5}}{4} \) = \(1.47\)