Arc length
Objective. Construct and evaluate Cartesian or parametric arc-length integrals.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
The arc length of a curve between two points is found by integrating the infinitesimal distance along the curve, using the formula L = ∫√(1+(dy/dx)²)dx for a Cartesian curve y=f(x), or L = ∫√((dx/dt)²+(dy/dt)²)dt for a curve given parametrically. This formula comes from approximating the curve by many tiny straight-line segments and applying Pythagoras' theorem to each, then taking the limit as an integral. For example, the arc length of y=x^(3/2) from x=0 to x=4 requires finding dy/dx=（3/2)x^(1/2) and substituting into the arc length formula.
Worked example
Find the arc length of the curve y = (2/3)x^(3/2) from x=0 to x=3.
- 1Differentiate: dy/dx = x^(1/2), so (dy/dx)² = x
- 2Arc length integral: L = ∫₀^3 √(1+x) dx
- 3Integrate: ∫√(1+x)dx = (2/3)(1+x)^(3/2), so L = [(2/3)(1+x)^(3/2)]₀^3
- 4Evaluate: (2/3)(4)^(3/2) − (2/3)(1)^(3/2) = (2/3)(8) − (2/3)(1) = 16/3 − 2/3 = 14/3
Answer: 14/3
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Practice
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Start practisingCommon mistake. Students forget the '1+' inside the square root, writing L=∫√((dy/dx)²)dx, which just integrates the gradient rather than the true arc length. Always include the 1 representing the horizontal component of each segment.
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