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Calculus/Arc Length of Parametric Curve

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Expert: - 6/9/2009


Question

Work on Integral
I am asked to find the arc length of the parametric curve x=2sin(t) &  y=cos(2t)  0<=t<=pi/3   

I am having some trouble solving the integral for this problem. I believe I have set it properly but the integral is giving me a hard time.  

The answer in the book is sqr(3)+(1/2) ln(sqr3+2)

MathCad produces an answer with an arcsinh in the answer.  Any tips on how to approach this integral. I very briefly covered my hyperbolic functions but still clueless.

Answer
x=2sin(t) & y=cos(2t) 0<=t<=pi/3

ds = sqrt( dx^2 + dy^2)

dx = 2 cos t dt
dy = - 2 sin(2t) dt

ds = 2 sqrt( cos^2(t) + sin^2(2t) ) dt

ds = 2 sqrt( cos^2(t) + 2 sin^2(t) cos^2(t)) dt

ds = 2 sqrt( 1 + 2 sin^2(t)) cos t dt

Now let  u = sin t:

ds = 2 sqrt( 1 + 2 u^2) du

Now that should be doable.

I see your stuff in the picture.  It's basically the same up to the u-substitution.

'The Integrator' web site gives the same arcsinh() result, BUT keep in mind that most of those can be done another way.  Try, perhaps:

let v = sqrt(2) u,  dv = sqrt(u) du

Then you will have:

ds = sqrt(2) sqrt( 1 + v^2) dv

and now a substitution of  v = tan z might do it.  

Paul Klarreich

Expertise

All topics in first-year calculus including infinite series, max-min and related rate problems. Also trigonometry and complex numbers, theory of equations, exponential and logarithmic functions. I can also try (but not guarantee) to answer questions on Analysis -- sequences, limits, continuity.

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I taught all mathematics subjects from elementary algebra to differential equations at a two-year college in New York City for 25 years.

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(See above.)

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