Question Number 11413 by agni5 last updated on 24/Mar/17

Find the length of the arc of the hyperbolic  spiral  rθ=a  lying between  r=a  and   r=2a.

Answered by mrW1 last updated on 26/Mar/17

r=(a/θ)  (dr/dθ)=−(a/θ^2 )  (√(r^2 +((dr/dθ))^2 ))=((a(√(1+θ^2 )))/θ^2 )  L=∫_θ_1  ^θ_2  (√(r^2 +((dr/dθ))^2 ))dθ=a∫_θ_1  ^θ_2  ((√(1+θ^2 ))/θ^2 )dθ  =a[−((√(1+θ^2 ))/θ)+ln (θ+(√(1+θ^2 )))]_θ_1  ^θ_2    =a[((√(1+θ_1 ^2 ))/θ_1 )−((√(1+θ_2 ^2 ))/θ_2 )+ln ((θ_2 +(√(1+θ_2 ^2 )))/(θ_1 +(√(1+θ_1 ^2 ))))]    with θ_1 =(a/r_1 )=(a/(2a))=(1/2) and θ_2 =(a/r_2 )=(a/a)=1  L=a[((√(1+(1/4)))/(1/2))−((√(1+1))/1)+ln ((1+(√(1+1)))/((1/2)+(√(1+(1/4)))))]  L=a[(√5)−(√2)+ln ((2(1+(√2)))/(1+(√5)))]

Commented bymrW1 last updated on 26/Mar/17

the answer is corrected.  please see also Q11433.