Calculating Arc Length by Writing As A Definite Integral

What is Arc Length?

Arc length refers to the distance along a curve between two points. Unlike straight-line distance, which is calculated using the Pythagorean theorem, arc length requires calculus to determine the exact length of a curved path. The concept is fundamental in physics, engineering, and computer graphics where precise measurements of curved paths are necessary.

For a curve defined by the function y = f(x) between points a and b, the arc length is the sum of all the infinitesimally small straight-line segments that approximate the curve. This sum becomes an integral when considering the limit as the segments become infinitely small.

The Formula

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

Where:

• L is the arc length
• f(x) is the function defining the curve
• dy/dx is the derivative of f(x) with respect to x
• a and b are the lower and upper limits of integration

This formula accounts for the fact that the length of each infinitesimal segment along the curve is slightly longer than the horizontal distance dx due to the vertical component dy.

How to Calculate Arc Length

Step 1: Define the Function

Start by identifying the function that describes your curve. For example, if you're working with a semicircle, you might use y = √(r² - x²) where r is the radius.

Step 2: Find the Derivative

Compute the derivative dy/dx of your function. This represents the slope of the curve at any point x. For the semicircle example, dy/dx = -x/√(r² - x²).

Step 3: Square the Derivative

Square the derivative to prepare for the integrand. In our example, (dy/dx)² = x²/(r² - x²).

Step 4: Form the Integrand

Add 1 to the squared derivative and take the square root. This gives the integrand √(1 + (dy/dx)²). For our semicircle, this becomes √(1 + x²/(r² - x²)) = r/√(r² - x²).

Step 5: Set Up the Integral

Integrate the simplified integrand from the lower limit a to the upper limit b. For a full semicircle, this would be from -r to r.

Step 6: Evaluate the Integral

Solve the definite integral to find the arc length. For the semicircle, the integral evaluates to πr, which matches the known circumference of a semicircle.

Worked Example

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

Step 1: Define the Function

f(x) = x²

Step 2: Find the Derivative

dy/dx = 2x

Step 3: Square the Derivative

(dy/dx)² = (2x)² = 4x²

Step 4: Form the Integrand

√(1 + (dy/dx)²) = √(1 + 4x²)

Step 5: Set Up the Integral

L = ∫₀¹ √(1 + 4x²) dx

Step 6: Evaluate the Integral

This integral doesn't have an elementary solution, so we would typically evaluate it numerically. Using a calculator or software, we find L ≈ 1.3607.

The exact value can be expressed in terms of elliptic integrals, but for practical purposes, the numerical approximation is sufficient.