Arc length measures the distance you would travel while moving along a curve, not just the straight-line distance between its endpoints. In calculus, many curves are easier to describe using a parameter or an angle instead of writing y directly as a function of x. Parametric and polar arc length formulas let us add up many tiny straight pieces of a curve to find its total length.
This matters in physics, engineering, robotics, and computer graphics whenever motion or shape follows a curved path.
For a parametric curve, the small distance traveled comes from the horizontal and vertical rates of change, dx/dt and dy/dt. For a polar curve, the distance depends on both how far the point is from the origin, r, and how fast that distance changes as the angle rotates, dr/dθ. In both cases, the formula has the same idea: integrate a speed-like quantity over the interval.
A correct setup requires matching the curve form, the parameter interval, and the derivative variables.
Key Facts
- Parametric arc length: L = ∫ from a to b sqrt((dx/dt)^2 + (dy/dt)^2) dt.
- Polar arc length: L = ∫ from α to β sqrt(r^2 + (dr/dθ)^2) dθ.
- For y = f(x), the arc length formula is L = ∫ from a to b sqrt(1 + (dy/dx)^2) dx.
- The parametric formula comes from speed: speed = ds/dt = sqrt((dx/dt)^2 + (dy/dt)^2).
- The polar formula comes from ds^2 = dr^2 + r^2 dθ^2, so ds/dθ = sqrt((dr/dθ)^2 + r^2).
- Example: x = t, y = t^2 on 0 ≤ t ≤ 1 gives L = ∫ from 0 to 1 sqrt(1 + 4t^2) dt.
Vocabulary
- Arc length
- The total distance measured along a curve between two specified points.
- Parametric curve
- A curve described by equations x = x(t) and y = y(t), where t is a parameter.
- Polar curve
- A curve described by r = r(θ), where r is distance from the origin and θ is the angle from the positive x-axis.
- Parameter
- An independent variable that traces a curve by controlling the values of x and y or another geometric quantity.
- Speed along a curve
- The rate at which position changes along a path, equal to ds/dt for a parametric curve.
Common Mistakes to Avoid
- Using only dy/dt in the parametric formula is wrong because arc length depends on both horizontal and vertical motion. Always include sqrt((dx/dt)^2 + (dy/dt)^2).
- Forgetting the r^2 term in the polar formula is wrong because changing angle moves the point along a circular direction even when r is constant. Use sqrt(r^2 + (dr/dθ)^2), not just |dr/dθ|.
- Mixing variables in the integral is wrong because the derivative and differential must match the curve description. If the curve uses t, integrate with dt; if it uses θ, integrate with dθ.
- Using endpoint coordinates as limits for a parametric or polar integral is wrong unless they are also the parameter or angle values. The limits must be the t-values or θ-values that trace the desired part of the curve.
Practice Questions
- 1 Find the arc length of the parametric curve x = 3t, y = 4t for 0 ≤ t ≤ 2.
- 2 Set up and evaluate the arc length of the polar curve r = 2 for 0 ≤ θ ≤ π.
- 3 A curve is given by x = cos t and y = sin t for 0 ≤ t ≤ 2π. Explain why the parametric arc length formula gives the circumference of a unit circle.