How do you find the length of the polar curve $r=cos^3(theta/3)$ ?

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Answer 1 · 2014-10-10T12:45:26

Use the chain rule.

By Chain Rule,

${dr}/{d theta}=3cos^2(theta/3)cdot[-sin(theta/3)]cdot1/3$

by cleaning up a bit,

$=-cos^2(theta/3)sin(theta/3)$

Let us first look at the curve $r=cos^3(theta/3)$ , which looks like this:

enter image source here

Note that $theta$ goes from $0$ to $3pi$ to complete the loop once.

Let us now find the length $L$ of the curve.

$L=int_0^{3pi}sqrt{r^2+({dr}/{d theta})^2} d theta$

$=int_0^{3pi}sqrt{cos^6(theta/3)+cos^4(theta/3)sin^2(theta/3)}d theta$

by pulling $cos^2(theta/3)$ out of the square-root,

$=int_0^{3pi}cos^2(theta/3)sqrt{cos^2(theta/3)+sin^2(theta/3)}d theta$

by $cos^2theta=1/2(1+cos2theta)$ and $cos^2theta+sin^2theta=1$ ,

$=1/2int_0^{3pi}[1+cos({2theta}/3)]d theta$

$=1/2[theta+3/2sin({2theta}/3)]_0^{3pi}$

$=1/2[3pi+0-(0+0)]={3pi}/2$

I hope that this was helpful.

How do you find the length of the polar curve r=cos^3(theta/3)r=cos^3(theta/3)?