Determining the Length of a Polar Curve

Key Questions

• How do you find the length of a petal of a polar curve?

  The Arc Length in Polar Coordinates is given bu:

  `L = int \ dS` where `dS=sqrt(r^2+((dr)/(d theta))^2) \ d theta`

  Steve M · 1 · Jan 3 2017
• How do you find the length of the polar curve `r=cos^3(theta/3)`?

  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 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.

  Wataru · 11 · Oct 10 2014
• How do you find the arc length of a polar curve?

  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`

  Wataru · · Sep 22 2014

• How do you find the arc length of a polar curve?
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• How do you find the length of a petal of a polar curve?
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• How do you find the exact length of the polar curve `r=1+sin(theta)` ?
• What is the arclength of `r=3/4theta` on `theta in [-pi,pi]`?
• What is the arclength of `r=4theta` on `theta in [-pi/4,pi]`?
• What is the arclength of `r=costheta-3sin2theta` on `theta in [-pi/4,pi]`?
• What is the arclength of `r=sin(theta^2-(7pi)/6) +2thetacos(theta -(3pi)/8)` on `theta in [(7pi)/8,pi]`?
• What is the arclength of `r=-sin(-5theta^2+(11pi)/8) +3thetacos(7theta -(5pi)/8)` on `theta in [(7pi)/8,(13pi)/8]`?
• What is the arclength of `r=-thetasin(theta/2-(27pi)/16) -3cos(theta/2 -(11pi)/16)` on `theta in [(11pi)/8,(13pi)/8]`?
• What is the arclength of `r=-cos(theta/2-(3pi)/8)/theta` on `theta in [(3pi)/4,(7pi)/4]`?
• What is the arclength of `r=-2sin(theta/4+(7pi)/8)` on `theta in [(pi)/4,(7pi)/4]`?
• What is the arclength of `r=-10sin(theta/4+(9pi)/8)` on `theta in [(pi)/4,(7pi)/4]`?
• What is the arclength of `r=-10sin(theta/4+(5pi)/16)` on `theta in [(-5pi)/16,(9pi)/16]`?
• What is the arclength of `r=-3cos(theta/16+(pi)/16)` on `theta in [(-5pi)/16,(9pi)/16]`?
• What is the arclength of `r=2sin(theta-(3pi)/16) + 9theta` on `theta in [(-5pi)/16,(9pi)/16]`?
• What is the arclength of `r=5/2sin(theta/8+(3pi)/16) -theta/4` on `theta in [(-3pi)/16,(7pi)/16]`?
• What is the arclength of `r=2sin(theta) -3theta` on `theta in [(-3pi)/8,(7pi)/8]`?
• What is the arclength of the polar curve `f(theta) = 2thetasin(5theta)-thetacot2theta` over `theta in [pi/12,pi/2]`?
• What is the arclength of the polar curve `f(theta) = 2thetasin(5theta)-thetacot2theta` over `theta in [pi/12,pi/3]`?
• What is the arclength of the polar curve `f(theta) = 2thetasin(5theta)-cottheta` over `theta in [pi/12,pi/6]`?
• What is the arclength of the polar curve `f(theta) = thetasin(3theta)-sec^2theta` over `theta in [0,pi/8]`?
• What is the arclength of the polar curve `f(theta) = 4sin(2theta)-2sec^2theta` over `theta in [0,pi/8]`?
• What is the arclength of the polar curve `f(theta) = -3sin(theta)-2sec^2theta` over `theta in [0,pi/8]`?
• What is the arclength of the polar curve `f(theta) = -3sin(3theta)-2cot4theta` over `theta in [0,pi/8]`?
• What is the arclength of the polar curve `f(theta) = sin(3theta)-4cot6theta` over `theta in [0,pi/4]`?
• What is the arclength of the polar curve `f(theta) = 2cos(9theta)-3sintheta` over `theta in [0,pi/3]`?
• What is the arclength of the polar curve `f(theta) = -5costheta+sin6theta` over `theta in [0,pi/3]`?
• What is the arclength of the polar curve `f(theta) = 2tan^2theta+cos3theta` over `theta in [0,(5pi)/12]`?
• What is the arclength of the polar curve `f(theta) = cos^2theta-3sin^2theta` over `theta in [pi/3,pi/2]`?
• What is the arclength of the polar curve `f(theta) = csc^2theta-cot^2theta` over `theta in [pi/3,pi/2]`?
• What is the arclength of the polar curve `f(theta) = sin4theta-cos3theta` over `theta in [pi/3,pi/2]`?
• What is the arc length of the polar curve `f(theta) = 5sintheta-4theta` over `theta in [pi/8, pi/3]`?
• What is the arc length of the polar curve `f(theta) = 2costheta-theta` over `theta in [pi/8, pi/3]`?
• What is the arc length of the polar curve `f(theta) = cos(3theta-pi/2) +thetacsc(-theta)` over `theta in [pi/12, pi/8]`?
• Question #d0a78
• Prove that the arc length of the polar curve `r = a(1-cos theta)` is `8a`?
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• Sketch the curve with the given polar equation...?