CK-12-Calculus

5.3. The Length of a Plane Curve http://www.ck12.org

Multimedia Links

The formula you just used to find the length of a curve was derived by using line segments to approximate the curve.
The derivation of that formula can be found at Wikipedia Entry on Arc Length. In the following applet you can
explore this further. Experiment with various curves and change the number of segments to see how changing the
number of segments is related to approximating the arc length. Arc Length Applet.
For video presentations showing how to obtain the arc length using parametric curves(16.0), see Just Math Tutori
ng, Arc Length Using Parametric Curves, Example 1 (8:17)

MEDIA

Click image to the left for use the URL below.

URL: http://www.ck12.org/flx/render/embeddedobject/574

and Just Math Tutoring, Arc Length Using Parametric Curves, Example 2 (7:27).

MEDIA

Click image to the left for use the URL below.

URL: http://www.ck12.org/flx/render/embeddedobject/575

Review Questions

1. Find the arc length of the curve

y=(x

(^2) + 2 ) 3 / 2
3
on[ 0 , 3 ].

2. Find the arc length of the curve

x=^16 y^3 + 21 y

ony∈[ 1 , 2 ].

3. Integrate

x=

∫y

0

√

sec^4 t− 1 dt,−π 4 ≤y≤π 4.

4. Find the length of the curve shown in the figure below. The shape of the graph is called theasteroidbecause
   it looks like a star. The equation of its graph isx^2 /^3 +y^2 /^3 = 1.