Find arc length of r =

Kyle Sutton 2022-07-04 Answered
Find arc length of r = 2 cos θ in the range 0 θ π ?
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Answers (2)

gozaderaradiox5
Answered 2022-07-05 Author has 19 answers
Explanation:
L = d s = θ 1 θ 2 r d θ
s = r θ so d s = r d θ , thus
2 0 π cos θ = 2 ( sin θ ) | 0 π = 2 ( sin π sin 0 ) = . . .
Considering that this is for a semicircle perimeter of a full cycle..
4 ( sin ] t h e t a ) | 0 π 2 = 4 ( sin ( π 2 ) sin 0 ) = 4 ( 1 0 ) = 4

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Audrina Jackson
Answered 2022-07-06 Author has 4 answers
It is easy to see that the curve is a circle of radius 1. It's length is obviously 2 π
A more analytic solution would go as follows
d s 2 = d r 2 + r 2 d θ 2
So, for r = 2 cos θ , we have
d r = 2 sin θ d θ
and hence
d s 2 = ( 2 sin θ d θ ) 2 + ( 2 cos θ ) 2 d θ 2 = 4 d θ 2
d s = 2 d θ
Thus, the arc length is
L = θ = 0 θ = π d s = 2 π

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