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???.
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}???
Let’s do another example.
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}???