(AD3) Arc Length#

In this lesson we are going to see how to:

  • calculate the length of a curve given in Cartesian coordinates.

Review Videos#

Arc Length

The length of the curve \(y=f(x)\) over the interval \(a\leq x \leq b\) is given by:

\[ \text{Length }= \int_a^b \sqrt{1+\left(\dfrac{dy}{dx}\right)^2} \, dx \]

provided the derivative, \(f'(x)\), is a continuous function on \([a,b]\).

Example 1#

Set up the integral that would give the length of the curve \(\quad y=\tfrac{1}{2}x^2 \quad \) between points \((0,0)\) and \((2,2)\).

Example 2#

Find the length of the curve \(\quad y=x^{3/2}\quad \) for \(1\leq x\leq 3\).

Example 3#

Find the length of the curve \(\quad x=y^2-\tfrac{1}{8}\ln y \quad \) for \(1\leq y\leq 5\).

Arc Length Formula#

Given a curve \(y=f(x)\) that lies between \(x=a\) and \(x=b\), to find the length we:

  1. Divide the interval \([a,b]\) into \(n\) subintervals.

  2. Use the endpoints to construct \(n\) line segments.

  3. Find the length of each line segment.

  4. Add the lengths together.

  5. Compute the limit as \(n\to \infty\).