Curve Lenght using Polar Coord.

[scott73]

Alright I am having a tough time with this problem, I am trying to find the length of a curve, r= theta^2 from Theta=0 to Theta=14,
I know that you need to use the equation
L= Integral from (0-14) of the squareroot of (theta^2)^2 + (2theta)^2,

I just dont know how to take the integral of this I guess, any help would be great
Scott

 

[soroban]

Hello, scott73!

Find the length of a curve, \(\displaystyle r\,=\,\theta^2\) from \(\displaystyle \theta = 0\) to \(\displaystyle \theta=14\)

I know that you need to use the equation:

\(\displaystyle \L L\;=\int^{\;\;\;\;14}_0\sqrt{(\theta^2)^2\,+\,(2\theta)^2}\, dx\)

You may kick yourself . . .


Simplify that radical: .\(\displaystyle \L\sqrt{\theta^4\,+\,4\theta^2}\:=\:\sqrt{\theta^2(\theta^2\,+\,4)}\:=\:\theta\sqrt{\theta^2\,+\,4}\)

And we have: .\(\displaystyle \L L\;=\int^{\;\;\;\;14}_0\theta(\theta^2\,+\,4)^{\frac{1}{2}}\,d\theta\)

Now let \(\displaystyle \L u\,=\,\theta^2\,+\,4\) . . . got it?

 

[scott73]

Um well, yeah, I did kick myself.... that was terrible, I was thinking so hard about the problem that I didnt even realize I could use a simple u substitution! I am embarased, thanks for the help!!!!!!