Parametrizing and Evaluating a Line Integral on a Given Curve

[meson0731]

Homework Statement

Evaluate ∫(x^3 + y^3)ds where C : r(t)=<e^t , e^(-t)>, 0 <= t <= ln2
c

Homework Equations

The Attempt at a Solution

I tried to parametrize the integral and change ds to sqrt(e^(2t) + e^(-2t)) dt.

I then change (x^3 + y^3) to (e^(3t) + e^(-3t)

so i ended up with


ln2
∫(e^(3t) + e^(-3t)) * sqrt(e^(2t) + e^(-2t))dt
0

I feel like i set the integral up wrong becuase I would have no idea of how to do this integral. Even wolframalpha gives me a crazy answer. Is there another way to do this or did i make a mistake?

 

[LCKurtz]

It looks correct to me. For what it's worth, Maple gives:$$
\frac 1 8\ln \left( \frac{13\sqrt{17}+51}{13\sqrt{17}-51}\right)+\frac{63\sqrt{17}}{64}$$

 

[meson0731]

So i have to actually do that integral? Is there a way to write it in differential form or another form that would be easier?

 

[LCKurtz]

So i have to actually do that integral? Is there a way to write it in differential form or another form that would be easier?

Beats me. I don't right off see a simple way to work it myself. I thought about expressing the integrand in terms of $\cosh(3t)$ and $\cosh(2t)$ and I still didn't see anything obvious. But then again, I haven't been losing any sleep over it and maybe someone else will see something clever.

 

[meson0731]

Yeah i tried changing it into hyperbolic but it just got even more messy...

 

[Villyer]

It may help to rewrite the curve that it travels over.
Instead of integrating $$r(t)=<e^t,e^{-t}>$$ it may be easier to integrate $$r(t)=<t,\frac{1}{t}>$$

It haven't tried it though, so it may not be any easier.

 

[nileszoso]

I'm working on this same problem. I did indeed rewrite it as r(t)=<t,1/t>, but this integral is no easier to solve. The solution given by Wolfram Alpha for this integral was the same numerically as that given by Maple in the above post.

 

[Villyer]

I haven't evaluated many line integrals, but if we are on the curve y = 1/x, can't we skip using the parameter t and integrate (x^3 + \frac{1}{x^3})\sqrt{1 + \frac{1}{x^4}} dx
from 1 to 2?
I'm having success evaluating it using the substitution x^2 = tan\theta.