Find the length of the curve over the given interval.

zeotropojd 2021-12-19 Answered
Find the length of the curve over the given interval. r=8+8cosθ on the interval 0θπ
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Expert Answer

Ronnie Schechter
Answered 2021-12-20 Author has 27 answers
Step 1
The given function is
r=8+8cosθ on the interval 0θπ
The length of the polar curve is given by
L=ab(r(θ))2+(r(θ))2dθ
First, find the derivative:
r(θ)(8cos(θ)+8)=8sin(θ)
Step 2
Finally, calculate the integral
L=0π(8cos(θ)+8)2+(8sin(θ))2dθ
=0π82cos(θ)+1dθ
=0π822cos2(θ2)dθ
=0π16cos(θ2)dθ
=32(sin(θ2))|{(θ=π)}32(sin(θ2))|(θ=0)
=32

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Nadine Salcido
Answered 2021-12-21 Author has 34 answers
Step 1
The definite integral that represents the arc length given by
Arc length =αβr2+(drdθ)2dθ
=0π(8+8cosθ)2+64sin2θdθ
=80π1+2cosθ+cos2θ+sin2θdθ
=80π2+2cosθdθ
=80π4cos2(θ2)dθ
=160πcos(θ2)dθ
=32sin(θ2)Big0π
=32

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