AP® Calculus BC · Unit 9 · Topic 9.3
Finding Arc Lengths of Curves Given by Parametric Equations
Parametric arc length adds up tiny straight-line distances traveled along a path, each one built from how fast x and y change together: integrate √((dx/dt)2 + (dy/dt)2) over t.
A parametric curve is traced by a point moving in the plane, with x and y each controlled by the same parameter t. At any instant, the point moves horizontally at rate dx/dt and vertically at rate dy/dt. These two rates are like the legs of a right triangle: the actual speed along the curve, the hypotenuse, is √((dx/dt)2 + (dy/dt)2). That's just the Pythagorean theorem applied to velocity. Multiply that speed by a tiny slice of time dt and you get a tiny piece of distance traveled. Add up all those tiny pieces from t = a to t = b with an integral, and you get the total arc length: L = integral from a to b of √((dx/dt)2 + (dy/dt)2) dt.
Working an example: for x = t2, y = t3, dx/dt = 2t and dy/dt = 3t2, so the integrand is √(4t2 + 9t4) = t*√(4+9t2) for t ≥ 0. Integrating from 0 to 1 gives a clean antiderivative via u-substitution (u = 4+9t2), producing a specific number. The cycloid x = t - sin(t), y = 1 - cos(t) works the same way: dx/dt = 1 - cos(t), dy/dt = sin(t), and after simplifying with the identity 1 - cos(t) = 2sin2(t/2), the integrand reduces to 2|sin(t/2)|, giving a total arch length of 8.
This formula is the direct descendant of the rectangular arc length formula from earlier work, and it sets up polar arc length next, since polar curves are just a special parametric form with x = r·cos(theta) and y = r·sin(theta).
Key terms
- Parametric curve
- A path in the plane where x and y are each given as separate functions of a parameter t.
- Speed along the path
- The magnitude √((dx/dt)2 + (dy/dt)2), combining horizontal and vertical rates of change.
- Arc length integral
- The integral from t=a to t=b of √((dx/dt)2 + (dy/dt)2) dt, giving total distance traveled.
- Cycloid
- The curve traced by a point on the rim of a rolling circle, given by x = t - sin(t), y = 1 - cos(t).
- Retracing
- When a curve passes over the same points more than once as t increases, inflating the arc length integral.
Watch it explained
- How it worksHow does the square root of (dx/dt)2 plus (dy/dt)2 become the speed along the path?
- Worked exampleFind the arc length of the curve x = t2, y = t3 from t = 0 to t = 1.
- Worked exampleFind the length of one arch of the cycloid x = t - sin(t), y = 1 - cos(t).
- Common mistakeWhy isn't parametric arc length the same as rectangular arc length when the path retraces itself?
Check yourself
End of topic.