- #1
mathnoobie
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Homework Statement
I was wondering if I did this problem correctly as I don't have the solution, also wanted to make sure that my limits of integration were correct as they tend to be tricky in finding arc length in polar coordinates.
x(t)=arcsint
y(t)=ln(sqrt(1-t^2))
Homework Equations
S= integral from a-b of
sqrt((dx/dt)^2+(dy/dt)^2)dt
The Attempt at a Solution
(dx/dt)^2=1/(1-t^2)
(dy/dt)^2=t^2/(1-t^2)^2
adding (dx/dt)^2+(dy/dt)^2
I get 1/(1-t^2)^2
Put all of this into the square root as said by the formula
I simplified it to the integral from 0 to 1/2 of dt/(1-t^2)
Factoring the bottom I get dt/((1-t)(1+t))
by Partial Fractions I get
2 separate integrals
(1/2)∫dt/(1-t)+(1/2)∫dt/(1+t)
Finally integrating this I get
(1/2)(ln(1-t)+ln(1+t))
Plugging in my limits of integration I get
(1/2)(ln(1/2)+ln(3/2))
Using the log rule
I get ln(3/4)^(1/2)
Thank you so much to anyone who read through this long problem!