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Finding the length of an arc given the vector?

M

Metalloid

Diamond Member
My last calc 3 hw problem, and I am completely stumped.

r(t) = <5e^(-5t), 5e^(5t), 25t*sqrt(2)>
I need to find the length of the curve over 2<t<5

So I differentiated each part of the vector, then squared them. Now I am supposed to add them together, take the square root, and somehow take the integral of that????

Of course the problems that they assign ALWAYS have e in them. Can someone please explain how to do this problem? Given, it's hard to type equations on this forum, but any insight at all would be much appreciated.

Thanks in advance.
 
Originally posted by: Metalloid
My last calc 3 hw problem, and I am completely stumped.

r(t) = <5e^(-5t), 5e^(5t), 25t*sqrt(2)>
I need to find the length of the curve over 2<t<5

So I differentiated each part of the vector, then squared them. Now I am supposed to add them together, take the square root, and somehow take the integral of that????

Of course the problems that they assign ALWAYS have e in them. Can someone please explain how to do this problem? Given, it's hard to type equations on this forum, but any insight at all would be much appreciated.

Thanks in advance.
Click to expand...

Yes, take the integral of that.
 
Have you tried reading the textbook?
I am pretty sure there's a specific integral formula which you just plug those vectors in.
 
Ok I can factor out the 625 (if you are working along with me)

That gives me 25 sqrt(e^-10t + e^10t +2)

I guess I'm not seeing how to integrate this. I can't do a U-sub. I can't integrate by parts. Any hints?
 
Originally posted by: Metalloid
Ok I can factor out the 625 (if you are working along with me)

That gives me 25 sqrt(e^-10t + e^10t +2)

I guess I'm not seeing how to integrate this. I can't do a U-sub. I can't integrate by parts. Any hints?
Click to expand...

Ok, I'll tell you.

(e^-5t + e^5t)^2 = e^-10t + e^10t +2
 
Originally posted by: chuckywang
Originally posted by: Metalloid
Ok I can factor out the 625 (if you are working along with me)

That gives me 25 sqrt(e^-10t + e^10t +2)

I guess I'm not seeing how to integrate this. I can't do a U-sub. I can't integrate by parts. Any hints?
Click to expand...

Ok, I'll tell you.

(e^-5t + e^5t)^2 = e^-10t + e^10t +2
Click to expand...

Ahhhh, e^5t multiplied by e^-5t is 1. Beautiful, I don't think I would have ever seen that.

Thanks for the help.
 
No no no, the length is equal to square root of the (dx squared PLUS dr squared). Think of the Pithagora
 
Originally posted by: Calin
No no no, the length is equal to square root of the (dx squared PLUS dr squared). Think of the Pithagora
Click to expand...

I'm not sure what you're trying to say. He's doing the problem correctly. Let's not confuse him more. 🙂
 
There is an explanation behind all those calculations: the length of the curve is the sum of the "elemental" curve elements on the entire interval. The length of an elemental curve is the hypotenuse of a triangle with one cathetus as dx and one cathetus as dy, or dr (as you named your function r).
This could helps if you forget the formulas
 
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