Length of Curve: Finding r(t) Derivative

[monnapomona]

Homework Statement

Find the length of the curve:

r(t) = e^-2t i + e^-2t*sin(t) j + e^-2t*cos(t) k, 0 ≤ t ≤ 2π

Homework Equations

L = ∫\stackrel{b}{a} |r'(t)|

The Attempt at a Solution

I tried factoring out e^-2t and got:

e^-2t (i + sin(t) j + cos(t) k)

This is where I got lost. Can I take the derivative this and just leave e^-2t outside of the brackets?

 

[haruspex]

Yes, differentiate the whole thing.

 

[monnapomona]

Yes, differentiate the whole thing.

Okay, would this right?

-2e^-2t (1 + cos(t) - sin(t))

 

[SammyS]

Homework Statement

Find the length of the curve:

r(t) = e^-2t i + e^-2t*sin(t) j + e^-2t*cos(t) k, 0 ≤ t ≤ 2π

Homework Equations

L = ∫\phantom{|}_{a}^{b} |r'(t)|

The Attempt at a Solution

I tried factoring out e^-2t and got:

e^-2t (i + sin(t) j + cos(t) k)

This is where I got lost. Can I take the derivative this and just leave e^-2t outside of the brackets?

Hello monnapomona. Welcome to PF !

e^-2t can be outside of the brackets, but can't be taken out of the integrand. But it may be better not to factor it out. Either way, you will need to use the product rule.

That integral needs the differential, dt:

\displaystyle \int_{0}^{2\pi}\left|\textbf{r}'(t)\right|\,dt​

What is \displaystyle \left|\textbf{r}'(t)\right|\ ?

 

[haruspex]

Okay, would this right?

-2e^-2t (1 + cos(t) - sin(t))

What's the derivative of f(t)g(t)?

 

[monnapomona]

What's the derivative of f(t)g(t)?

I think it would be f'(t)g(t) + g'(t)f(t).

I tried not factoring out e^-2t from the vector function and got this

r'(t) = e^-2t - 2e^-2tsin(t) +e^-2tcos(t) - e^-2tsin(t) - 2e^-2tcos(t)


I'm lost on what to do next... :s

 

[SammyS]

I think it would be f'(t)g(t) + g'(t)f(t).

I tried not factoring out e^-2t from the vector function and got this

r'(t) = e^-2t - 2e^-2tsin(t) +e^-2tcos(t) - e^-2tsin(t) - 2e^-2tcos(t)

I'm lost on what to do next... :s

Where are the unit vectors?

 

[monnapomona]

Where are the unit vectors?

Oh, whoops.

r'(t) = e^-2t i - 2e^-2tsin(t)+e^-2tcos(t) j - e^-2tsin(t) - 2e^-2tcos(t) k

I'm not sure if this is okay to do but I tried factoring out the e^-2t and got

r'(t) = e^-2t ( i - 2sin(t) + cos(t) j - sin(t) - 2cos(t) k )

 

[aralbrec]

It looks like the i part is wrong so you may want to doublecheck.

But I'm not sure why you stopped there :) Length of a vector is...

 

[monnapomona]

Where are the unit vectors?

It looks like the i part is wrong so you may want to doublecheck.

But I'm not sure why you stopped there :) Length of a vector is...

Oh yeah! okay hopefully this is right:

r'(t) = -2e^-2t i - 2e^-2tsin(t)+e^-2tcos(t) j - e^-2tsin(t) - 2e^-2tcos(t) k

i'm a little confused on how to do length...
Could I square each vector like (i, j, k) and just add what I get for it together?

 

[aralbrec]

Oh yeah! okay hopefully this is right:

r'(t) = -2e^-2t i - 2e^-2tsin(t)+e^-2tcos(t) j - e^-2tsin(t) - 2e^-2tcos(t) k

That looks right. I would have kept the exponential factored out to make it easier to take the magnitude.

i'm a little confused on how to do length...
Could I square each vector like (i, j, k) and just add what I get for it together?

Yes and take square root. This vector is no different than any other. It has a direction and magnitude; the only difference from what you are used to is maybe that its direction and magnitude change with t (time).

For (2d) vectors v = ai + bj, |v| = √(a² + b²). Sometimes it is more convenient to use the dot product: v.v = a² + b², |v| = √(v.v)

Sometimes it's not always taught where this length integral comes from. You should notice that if r(t) is a position vector, then r'(t) is velocity, which makes speed |r'(t)|. Since distance = rate * time, the distance traveled in time dt is |r'(t)| dt. Add up the distances traveled during the entire time interval and you get the distance travelled.

 

[monnapomona]

That looks right. I would have kept the exponential factored out to make it easier to take the magnitude.
Yes and take square root. This vector is no different than any other. It has a direction and magnitude; the only difference from what you are used to is maybe that its direction and magnitude change with t (time).

For (2d) vectors v = ai + bj, |v| = √(a² + b²). Sometimes it is more convenient to use the dot product: v.v = a² + b², |v| = √(v.v)

Sometimes it's not always taught where this length integral comes from. You should notice that if r(t) is a position vector, then r'(t) is velocity, which makes speed |r'(t)|. Since distance = rate * time, the distance traveled in time dt is |r'(t)| dt. Add up the distances traveled during the entire time interval and you get the distance travelled.

Okay, I took the derivative of all the components and my length formula turned out to look like this:

L = ∫\stackrel{2\pi}{0} √(4e^-4t + e^-4tsin²t + 4e^-4tcos²t + e^-4tcos²t + 4e^-4tsin²t)

= e^-2t√( 4 + 5(cos²t + sin²t )
= e^-2t√(4+5)
= 3e^-2t

Using the fundamental theorem of calc, I got the answer to be 3e^-4π - 3 ? Would this be the right answer?

 

[SammyS]

Oh yeah! okay hopefully this is right:

r'(t) = -2e^-2t i - 2e^-2tsin(t)+e^-2tcos(t) j - e^-2tsin(t) - 2e^-2tcos(t) k

i'm a little confused on how to do length...
Could I square each vector like (i, j, k) and just add what I get for it together?

You really need extra parentheses:

r'(t) = -2e^-2t i + (- 2e^-2tsin(t)+e^-2tcos(t) ) j + (- e^-2tsin(t) - 2e^-2tcos(t) )k​

Then, in you last post it appears that your integration isn't quite right.

 

[monnapomona]

You really need extra parentheses:

r'(t) = -2e^-2t i + (- 2e^-2tsin(t)+e^-2tcos(t) ) j + (- e^-2tsin(t) - 2e^-2tcos(t) )k​

Then, in you last post it appears that your integration isn't quite right.

Oh wow, I forgot to integrate it... I got this as the answer:

\inte^-2t = -1/2 e^-2t

so -3/2(e^(-2t)) | ^{2π}_{0}

= -3/2(e^-4π + 1)

 

[SammyS]

Oh wow, I forgot to integrate it... I got this as the answer:

\inte^-2t = -1/2 e^-2t

so -3/2(e^(-2t)) | ^{2π}_{0}

= -3/2(e^-4π + 1)

Almost ...

It's -3/2(e^-4π - 1)

 

[monnapomona]

Almost ...

It's -3/2(e^-4π - 1)

Oh I see where I went wrong.

Thank you for all your help!