perimeter

Raju

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Sep 2, 2009
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Please help in solving

Find the perimeter of the cord r = a (1+cos ?)

Note:
? means "Theeta".
 
Raju said:
Please help in solving

Find the perimeter of the cord r = a (1+cos ?)

Note:
? means "Theeta".

First define the problem fully!

Is it a circle you are talking about?

What is 'r' and 'a' and 'theta'?

Where are these located (is there a picture)?
 
Hello, Raju!

Find the perimeter of the cardioid:   r=a(1+cosθ)\displaystyle \text{Find the perimeter of the cardioid: }\;r \:=\:a(1+\cos\theta)

Arc length for polar functions:   L  =  αβr2+(drdθ)2dθ\displaystyle \text{Arc length for polar functions: }\;L \;=\;\int^{\beta}_{\alpha} \sqrt{r^2 + \left(\tfrac{dr}{d\theta}\right)^2}\,d\theta

We have:   r=a(1+cosθ)drdθ=asinθ\displaystyle \text{We have: }\;r \:=\:a(1+\cos\theta) \qquad \tfrac{dr}{d\theta} \:=\:-a\sin\theta

Then:   r2+(drdθ)2  =  [a(1+cosθ)]2+[asinθ]2  =  a2(1+2cosθ+cos2 ⁣θ)+a2sin2θ\displaystyle \text{Then: }\;r^2 + \left(\tfrac{dr}{d\theta}\right)^2 \;=\;\bigg[a(1+\cos\theta)\bigg]^2 + \bigg[-a\sin\theta\bigg]^2 \;=\; a^2(1 + 2\cos\theta + \cos^2\!\theta) + a^2\sin^2\theta

. . =  a2[1+2cosθ+cos2 ⁣θ+sin2 ⁣θ]  =  a2[2+2cosθ]  =  2a2(1+cosθ)\displaystyle = \;a^2\bigg[1 + 2\cos\theta + \underbrace{\cos^2\!\theta + \sin^2\!\theta} \bigg] \;=\;a^2[2 + 2\cos\theta] \;=\;2a^2(1 + \cos\theta)
. . . . . . . . . . . . . . . . . This is 1\displaystyle ^{\text{This is 1}}

. . =  4a21+cosθ2  =  4a2cos2 ⁣θ2\displaystyle = \;4a^2\cdot\frac{1+\cos\theta}{2} \;=\; 4a^2\cos^2\!\tfrac{\theta}{2}

Hence:   r2+(drdθ)2  =  4a2cos2 ⁣θ2  =  2acos ⁣θ2\displaystyle \text{Hence: }\;\sqrt{r^2 + \left(\tfrac{dr}{d\theta}\right)^2} \;=\;\sqrt{4a^2\cos^2\!\tfrac{\theta}{2}} \;=\; 2a\cos\!\tfrac{\theta}{2}



Due to the symmetry, we can find the length from 0 to π and multiply by 2.\displaystyle \text{Due to the symmetry, we can find the length from 0 to }\pi\text{ and multiply by 2.}

. . Therefore:   L  =  2×2a0πcos ⁣θ2dθ\displaystyle \text{Therefore: }\;L \;=\;2 \times 2a\int^{\pi}_0 \cos\!\tfrac{\theta}{2}\,d\theta

Go for it!

 
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