Back to Parametric Equations, Polar Coordinates, and Vector-Valued Functions

Lesson PreviewLesson 3 of 9 in this unit

9.3 Finding Arc Lengths of Curves Given by Parametric Equations

Part of Parametric Equations, Polar Coordinates, and Vector-Valued Functions in AP® Calculus BC. Read the opening section, then create a free account for the full interactive lesson, practice, and progress tracking.

Start free
If a curve is given parametrically by x=f(t) and y=g(t) for a≤t≤b{a\le t\le b}, then its arc length is
L=∫ab(dxdt)2+(dydt)2 dtL=\int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}\,dt
This formula comes from treating a tiny piece of the curve like a tiny line segment and applying the Pythagorean Theorem. Because both x and y change with t, the total path length depends on both rates together.

You should think of this as the parametric version of the arc-length formula from rectangular coordinates. Instead of using f'(x), you use both coordinate derivatives. The expression under the square root measures how fast the particle or point is moving through the plane. If that expression is easy to integrate, you can find an exact arc length; otherwise, the integral may be left in exact form.

In this unit

9 lessons

  1. 9.1Defining and Differentiating Parametric EquationsPreview lesson
  2. 9.2Second Derivatives of Parametric EquationsPreview lesson
  3. 9.3Finding Arc Lengths of Curves Given by Parametric EquationsThis lesson
  4. 9.4Defining and Differentiating Vector-Valued FunctionsPreview lesson
  5. 9.5Integrating Vector-Valued FunctionsPreview lesson
  6. 9.6Solving Motion Problems Using Parametric and Vector-Valued FunctionsPreview lesson
  7. 9.7Defining Polar Coordinates and Differentiating in Polar FormPreview lesson
  8. 9.8Find the Area of a Polar Region or the Area Bounded by a Single Polar CurvePreview lesson
  9. 9.9Finding the Area of the Region Bounded by Two Polar CurvesPreview lesson

Continue the full lesson in Stellar

Create a free account to unlock the remaining sections, checkpoints, course practice, and progress tracking for AP® Calculus BC.

Start free