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Find the exact length of the curve

Calculus
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Find the exact length of the curve

x = et + ety = 5 − 2t,    0 ≤ t ≤ 2 

Apr 3rd, 2015

x = e^t + e^-t

dx = (e^t - e^-t)dt

 dx² = e^(2t) -2 + e^(-2t) 
y = 5 -2t,

 dy = -2dt

 dy² = 4 
dx² + dy² = e^(2t) -2 + e^(-2t) + 4

 = e^(2t) +2 + e^(-2t)

= (e^t + e^-t)² 
the length is given by: 
s = ∫(dx² + dy²)^(1/2)dt

 = ∫[(e^t + e^-t)²]^(1/2)dt

 =∫(e^t + e^-t)dt 
s = (e^t - e^-t) (0 to 2)

= (e^2 - e^0)-(e^-2 - e^0)

= 7.2537


Apr 3rd, 2015

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