Interval for the Length of an Arc

In summary: so you would do it like this:you would differentiate and pull out a common factor and simplify the sin and cos to 1. and then add the extra 1 and take the square root.
  • #1
ineedhelpnow
651
0
find the length of an arc of a helix r(t)=(sint,cost,t) from the point (0,2,0) to (0,5,2pi)

would the interval when integrating be from 0 to 2pi because t in the case is (z=t)? please say yes. please say yes.
 
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  • #2
ineedhelpnow said:
find the length of an arc of a helix r(t)=(sin2t,cos2t,t) from the point (0,2,0) to (0,5,2pi)

would the interval when integrating be from 0 to 2pi because t in the case is (z=t)? please say yes. please say yes.

Yes.
 
  • #3

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  • #4
ineedhelpnow said:
really? is this right?

It would be right... but I'm only just now noticing that your function is not a helix.
How does it get to $y=5$? (Wondering)
 
  • #5
ok ILS ima let you in on a little secret (i don't really remember the points. all i remember are the z coordinates 0 and 2pi) but it was a helix for sure. it was probably 5 instead of 2. i don't know i can't remember. i don't want to say i made them up but I am going to saaaaay improvised.
 
  • #6
ineedhelpnow said:
ok ILS ima let you in on a little secret (i don't really remember the points. all i remember are the z coordinates 0 and 2pi)

Okay...
How about $r(t)=\left(\frac{3t}{2\pi}\sin(t),\ 2+\frac{3t}{2\pi}\cos(t),\ t\right)$?
 
  • #7
what about it?
you would differentiate and pull out a common factor and simplify the sin and cos to 1. and then add the extra 1 and take the square root.
 
  • #8
ineedhelpnow said:
what about it?
you would differentiate and pull out a common factor and simplify the sin and cos to 1. and then add the extra 1 and take the square root.

Did you really differentiate it?
How about the factor $t$ that is in both the x-coordinate and the y-coordinate (which is an integral part of a helix)?
 
  • #9
factor it out before you differentiate.
 
  • #10
ineedhelpnow said:
factor it out before you differentiate.

Can you show that?

Edit: Perhaps I should say: you can't.
 
  • #11
why can't it be factored out? :confused:
 
  • #12
ineedhelpnow said:
why can't it be factored out? :confused:

I'm not entirely sure what you mean, but I'm going out on a limb and say: no, it can't be factored out.

What is the derivative of $t \cos t$?
 
  • #13
cos t - t*sin t
 
  • #14
ineedhelpnow said:
cos t - t*sin t

There you go!
No factoring out before the differentiation.
 
  • #15
ineedhelpnow said:
what about it?
you would differentiate and pull out a common factor and simplify the sin and cos to 1. and then add the extra 1 and take the square root.
:D soooo you can just do what i said earlier. by differentiating. pulling out a common factor of 3/2pi and---- oh i see what i did wrong. (Giggle) made a mistake with the differentiating part.
 

FAQ: Interval for the Length of an Arc

What is an interval for the length of an arc?

An interval for the length of an arc is a range of values that represents all possible lengths of an arc within a given circle or curve.

How is an interval for the length of an arc calculated?

The interval for the length of an arc is calculated using the formula L = rθ, where L is the length of the arc, r is the radius of the circle or curve, and θ is the central angle in radians.

What is the significance of an interval for the length of an arc?

An interval for the length of an arc allows us to determine the possible lengths of an arc within a given circle or curve, providing a more accurate measurement than a single value.

Can an interval for the length of an arc be negative?

No, an interval for the length of an arc cannot be negative as it represents a range of positive values for the length of the arc.

How can an interval for the length of an arc be used in real-world applications?

An interval for the length of an arc can be used in various fields such as engineering, architecture, and physics to accurately determine the length of a curved object or measure the distance between two points on a circular path.

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