- #1

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## Main Question or Discussion Point

I have a curve defined by following parametric equation:

\begin{equation}

\gamma(\theta) = 1 + 0.5 \times \cos (N \theta) (\cos(\theta),\sin(\theta)), 0 \leq \theta \leq 2 \pi \

\end{equation}

I need to calculate the length of the curve between say θ = 0 to θ = 1.0

Formula for calculating the arc length of a curve in polar form is well known (see http://mathworld.wolfram.com/ArcLength.html Eq. 17 therein). Using this formula

\begin{equation}

s = \int\limits_{\theta = 0}^{\theta = 1.0} \sqrt{r^2 +\left(\frac{dr}{d\theta}\right)^2}

\end{equation}

where for N = 10, r is given as

\begin{equation}

r = 1 + 0.5 \times \cos(10 \theta)

\end{equation}

and

\begin{equation}

\frac{dr}{d\theta} = -0.5 \times \sin(10\theta)

\end{equation}

The problem is I can't evaluate the integral in Eq. 17 from the Mathworks link for the above equation explicitly. Obviously one can use Matlab ( with trapz command ) to evaluate the integral for given limits but I want an explicit expression for the indefinite integral in terms of generic θ1 and θ2 ( I have hundreds of such integrals to evaluate and difference between θ1 and θ2 is not constant) How do I evaluate the integral? Is there a substitution possible?

Many thanks for help.

\begin{equation}

\gamma(\theta) = 1 + 0.5 \times \cos (N \theta) (\cos(\theta),\sin(\theta)), 0 \leq \theta \leq 2 \pi \

\end{equation}

I need to calculate the length of the curve between say θ = 0 to θ = 1.0

Formula for calculating the arc length of a curve in polar form is well known (see http://mathworld.wolfram.com/ArcLength.html Eq. 17 therein). Using this formula

\begin{equation}

s = \int\limits_{\theta = 0}^{\theta = 1.0} \sqrt{r^2 +\left(\frac{dr}{d\theta}\right)^2}

\end{equation}

where for N = 10, r is given as

\begin{equation}

r = 1 + 0.5 \times \cos(10 \theta)

\end{equation}

and

\begin{equation}

\frac{dr}{d\theta} = -0.5 \times \sin(10\theta)

\end{equation}

The problem is I can't evaluate the integral in Eq. 17 from the Mathworks link for the above equation explicitly. Obviously one can use Matlab ( with trapz command ) to evaluate the integral for given limits but I want an explicit expression for the indefinite integral in terms of generic θ1 and θ2 ( I have hundreds of such integrals to evaluate and difference between θ1 and θ2 is not constant) How do I evaluate the integral? Is there a substitution possible?

Many thanks for help.