How to find the arc length of a polar curve

The formula we use to find the arc length of a polar curve

The arc length of a polar curve is simply the length of a section of a polar curve between two points $a$ and $b$.

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We use the formula

$L=\int^b_a\sqrt{r^2+\left(\frac{dr}{d\theta}\right)^2}d\theta$

where $L$ is the arc length

where $r$ is the equation of the polar curve

where $\frac{dr}{d\theta}$ is the derivative of the polar curve

where $a$ and $b$ are the endpoints of the section

How to find the arc length of a curve given in polar coordinates

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Let’s do a couple examples where we find the arc length of a polar curve over a particular interval

Example

Find the arc length of the polar curve over the given interval.

$r=\cos^2{\frac{\theta}{2}}$

$0\le\theta\le\frac{\pi}{2}$

Before we can plug into the arc length formula, we need to find $dr/d\theta$.

$\frac{dr}{d\theta}=2\cos{\frac{\theta}{2}}\left[-\sin{\frac{\theta}{2}}\right]\left(\frac12\right)$

$\frac{dr}{d\theta}=-\cos{\frac{\theta}{2}}\sin{\frac{\theta}{2}}$

Now we can go ahead and solve for the arc length

$L=\int^{\frac{\pi}{2}}_0\sqrt{\left(\cos^2{\frac{\theta}{2}}\right)^2+\left(-\cos{\frac{\theta}{2}}\sin{\frac{\theta}{2}}\right)^2}\ d\theta$

$L=\int^{\frac{\pi}{2}}_0\sqrt{\cos^4{\frac{\theta}{2}}+\cos^2{\frac{\theta}{2}}\sin^2{\frac{\theta}{2}}}\ d\theta$

$L=\int^{\frac{\pi}{2}}_0\sqrt{\cos^2{\frac{\theta}{2}}\left(\cos^2{\frac{\theta}{2}}+\sin^2{\frac{\theta}{2}}\right)}\ d\theta$

Since $\cos^2{x}+\sin^2{x}=1$, we get

$L=\int^{\frac{\pi}{2}}_0\sqrt{\left(\cos^2{\frac{\theta}{2}}\right)(1)}\ d\theta$

$L=\int^{\frac{\pi}{2}}_0\cos{\frac{\theta}{2}}\ d\theta$

$L=2\sin{\frac{\theta}{2}}\Big|^{\frac{\pi}{2}}_0$

$L=2\sin{\left(\frac{\frac{\pi}{2}}{2}\right)}-2\sin{\left(\frac{0}{2}\right)}$

$L=2\sin{\frac{\pi}{4}}-2\sin{0}$

$L=2\cdot\frac{\sqrt{2}}{2}-2(0)$

$L=\sqrt{2}$

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Let’s do another example.

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The arc length of a polar curve is simply the length of a section of a polar parametric curve between two points a and b.

Example

Find the arc length of the polar curve over the given interval.

$r=e^{2\theta}$

$0\le\theta\le\pi$

Before we can plug into the arc length formula, we need to find $dr/d\theta$.

$\frac{dr}{d\theta}=2e^{2\theta}$

Plugging everything into the formula, we get

$s=\int^{\pi}_0\sqrt{\left(e^{2\theta}\right)^2+\left(2e^{2\theta}\right)^2}\ d\theta$

$s=\int^{\pi}_0\sqrt{e^{4\theta}+4e^{4\theta}}\ d\theta$

$s=\int^{\pi}_0\sqrt{5e^{4\theta}}\ d\theta$

$s=\sqrt{5}\int^{\pi}_0e^{2\theta}\ d\theta$

$s=\frac{\sqrt{5}}{2}e^{2\theta}\bigg|^{\pi}_0$

$s=\frac{\sqrt{5}}{2}e^{2(\pi)}-\frac{\sqrt{5}}{2}e^{2(0)}$

$s=\frac{\sqrt{5}}{2}e^{2\pi}-\frac{\sqrt{5}}{2}$

$s=\frac{\sqrt{5}\left(e^{2\pi}-1\right)}{2}$

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