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C39 · Unit C9 Parametric, polar and vector calculus

Parametric arc length and vector derivatives

BC extensionReview editionNot yet human-reviewed
Time
About 45 minutes, untimed
Course framework reference
AP 9.3–9.4 · BC
Builds on
Part of

Progress on this lesson counts in every pathway listed.

Understand the idea

A plane curve r(t) equals 〈x(t),y(t)〉 has velocity 〈x minutes,y minutes 〉 and speed square root of (x minutes squared +y minutes squared). Its length on [a,b] is the integral of speed. Differentiate vector components independently. A retraced curve contributes its traveled length every time it is traversed; it is not just the length of the geometric set.

Worked example

For r(t) equals 〈3cos t,3sin t〉, velocity is 〈 minus 3sin t,3cos t〉 and speed is 3. From t equals 0 to 2 pi the path length is 6 pi; from 0 to 4 pi it is 12 pi because the circle is traced twice.

Explore the mathematics

Parametric arc length and vector derivatives

Find speed from two coordinate derivatives, then integrate path length.

Setting 3 of 5

Scroll sideways to see the full diagram.

How does a parameter’s rate turn into speed along a path?xy-2-112x=cos(t), y=sin(t); 0≤t≤3π/12

Predict before continuing

How does a parameter’s rate turn into speed along a path?

Read the complete explanation
  1. How does a parameter’s rate turn into speed along a path?
  2. Differentiate both coordinates: x′=−sin(t), y′=cos(t).
  3. Speed=√(x′²+y′²)=√(sin²t+cos²t)=1.
  4. Arc length is ∫1 dt=0.7854. This is why radians equal arc length divided by radius for the unit circle.
Parametric arc length and vector derivatives · visual worked method
Parametric arc length and vector derivatives · visual worked methodFor r(t)=〈3cos t,3sin t〉, velocity is 〈−3sin t,3cos t〉 and speed is 3. From t=0 to 2π the path length is 6π; from 0 to 4π it is 12π because the circle is traced twice.Worked example · 1/2For r(t)=〈3cos t,3sin t〉, velocityis 〈−3sin t,3cos t〉 and speed is3. From t=0 to 2π the path lengthis 6π; from 0 to 4π itWorked example · 2/2is 12π because the circle istraced twice.

For r(t)=〈3cos t,3sin t〉, velocity is 〈−3sin t,3cos t〉 and speed is 3. From t=0 to 2π the path length is 6π; from 0 to 4π it is 12π because the circle is traced twice.

Discuss: explain what each label represents, then draw a related example with different values.

A common misconception

Integrating a vector gives displacement. Integrating its magnitude gives length; those operations are not interchangeable.

Investigate and explain

Trace the same circle once slowly and once quickly. Compare instantaneous speeds, total displacement and total path length.
Identify one step in the worked example, name the mathematical fact that permits it, and state its domain conditions.