Arc Length of Polar Curves
For polar curves of the form rθ, the arc length of a curve on the interval a ≤ θ ≤ b can be calculated using an integral.
Calculating Arc Length
The x- and y-coordinates of any Cartesian point can be written as the following parametric equations:
x=r cosθy = r sinθ a≤ θ ≤b.
Using the chain rule, note that dxdθ = drdθ cosθ − r sinθ and dydθ = drdθ sinθ + r cosθ.
For these parametric equations, the arc length of a curve on the interval a ≤ θ ≤ b is given by:
L = ∫ab dxdθ2 + dydθ2 ⅆθ
L =∫ab drdθ cosθ − r sinθ2 + drdθ sinθ + r cosθ2 ⅆθ
L =∫ab drdθ2cos2θ − 2 r drdθcosθ sinθ + r2 sin2θ + drdθ2sin2θ + 2 r drdθcosθ sinθ + r2 cos2θ ⅆθ
L =∫ab drdθ2cos2θ + sin2θ+ r2 sin2θ +cos2θ ⅆθ
Remembering that cos2θ + sin2θ = 1, this expression can now be simplified to:
L =∫ab drdθ2+ r2 ⅆθ
Therefore the arc length for a polar curve on the interval a ≤ θ ≤ b is L =∫ab r2 + drdθ2 ⅆθ.
Choose a polar function from the list below to plot its graph. Enter the endpoints of an interval, then use the slider or button to calculate and visualize the arc length on the given interval. When choosing the endpoints, remember to enter π as "Pi".
rθ = 1thetasqrt(theta + 1)cos(theta)1 - sin(theta)cos(2*theta)sin(3*theta)cos(theta) + sin(theta)3*cos(4*theta)2 - 4*cos(theta)5 + sin(7*theta)cos(sqrt(theta))exp(cos(theta/4))exp(cos(4*theta))theta^sin(theta)sqrt(sin(theta)^2)
Interval θ =
The arc length of the curve on this interval is
Download Help Document
What kind of issue would you like to report? (Optional)