Determining the Length of a Polar Curve
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Key Questions

We can find the arc length
#L# of a polar curve#r=r(theta)# from#theta=a# to#theta=b# by#L=int_a^bsqrt{r^2+({dr}/{d theta})^2}d theta# 
The Arc Length in Polar Coordinates is given bu:
# L = int \ dS # where# dS=sqrt(r^2+((dr)/(d theta))^2) \ d theta # 
Answer:
Use the chain rule.
Explanation:
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: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 squareroot,#=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.