AP Calculus BC 9.3 Finding Arc Lengths of Curves Given by Parametric Equations Study Notes - New Syllabus

 Finding Arc Lengths of Curves Given by Parametric Equations

In many real-world situations, curves are not expressed as a single equation \( y = f(x) \), but instead are given in parametric form where both \( x \) and \( y \) are defined in terms of a parameter \( t \). For example, a particle moving along a path in the plane may have its horizontal and vertical positions described by \( x(t) \) and \( y(t) \).

The arc length of a curve is the total distance traveled along the curve between two points. For parametric curves, we extend the basic idea of arc length from Cartesian equations by applying the Pythagorean theorem to small changes in \( x \) and \( y \).

Derivation of the Formula:

For a small change in parameter \( t \), the horizontal change is

\( dx = \dfrac{dx}{dt} \, dt \) and the vertical change is \( dy = \dfrac{dy}{dt} \, dt \).

By the Pythagorean theorem, the small segment length \( ds \) is given by:

$ ds = \sqrt{(dx)^{2} + (dy)^{2}} = \sqrt{\left( \dfrac{dx}{dt} \right)^{2} + \left( \dfrac{dy}{dt} \right)^{2}} \, dt $

To find the total length between \( t = a \) and \( t = b \), integrate:

$ L = \displaystyle \int_{a}^{b} \sqrt{\left( \dfrac{dx}{dt} \right)^{2} + \left( \dfrac{dy}{dt} \right)^{2}} \, dt $

Key Points to Remember:

• The formula works for any smooth parametric curve where \( \dfrac{dx}{dt} \) and \( \dfrac{dy}{dt} \) are continuous on \([a, b]\).
• The parameter \( t \) does not have to represent time  it is just a way of tracing out the curve.
• The integrand under the square root is the speed of motion along the curve.
• If \( x \) and \( y \) are given as functions of \( \theta \) (such as in polar coordinates), the same formula applies with \( t \) replaced by \( \theta \).
• Sometimes the integral can be simplified by factoring, trigonometric identities, or substitutions.