Length of Curve (Calc 2) question

In summary, the length of a curve given by a vector r is L = \int |r'(t)|dt. If the vector is given by r(t) = <t, t^2/sqrt2, t^3/3> on an interval from 0 to 1, the length can be calculated by setting up the integral using the magnitude of the derivative of r(t) and solving for L = \int (1^2 + (2t/sqrt2)^2 + (t^2)^2)dt. The t value can be set equal to something to make the magnitude work for an integral that would result in integrating dt. The final equation for L is L = \int \limits_a^b
  • #1
twiztidmxcn
43
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So the length of a curve given by a vector r is:

L = [tex] \int |r'(t)|dt[/tex]

If the vector is given by r(t) = < t, t^2/srt2, t^3/3> on an interval from 0 to 1, find the length

I am incredibly confused here, as most of the examples we've used have been using the cos^2+sin^2 = 1 identity.

I took the derivative of r(t), and got

r'(t) = 1, 2t/sqrt2, t^2>

then i set up the integral using the magnitude of that equation, and end up with L = [tex] \int (1^2 + (2t/sqrt2)^2 + (t^2)^2)dt [/tex]

I get this feeling like I need to set the t value equal to something in order to make the magnitude work out for an integral that would work out to integrating something like dt itself.

Any help would be much appreciated
Thank you
Twiztidmxcn
 
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  • #2
You forgot the squareroot and the limits, but the rest looks fine. You just have to calculate

[tex] L = \int \limits_a^b | r'(t) |\mathrm{d} t = \int \limits_0^1 \sqrt{1^2 + (\sqrt{2}t)^2 + (t^2)^2 } \mathrm{d} t [/tex]
 
  • #3
Just to help you a bit further:
Note that:
[tex]1^{2}+(\sqrt{2}t)^{2}+(t^{2})^{2}=1+2t^{2}+t^{4}=(1+t^{2})^{2}[/tex]
 

FAQ: Length of Curve (Calc 2) question

What is the length of a curve?

The length of a curve is the distance along the curve from one end point to the other. It is calculated by finding the integral of the curve's derivative.

How do you find the length of a curve?

To find the length of a curve, you must first find the derivative of the curve. Then, you can integrate the derivative over the interval of the curve to find the length.

What is the formula for finding the length of a curve?

The formula for finding the length of a curve is ∫√(1 + (dy/dx)^2) dx, where dy/dx is the derivative of the curve.

Why is finding the length of a curve important?

Finding the length of a curve is important in various fields, such as physics and engineering, where understanding the length of a path or trajectory is crucial for calculations and predictions.

What are the main challenges in finding the length of a curve?

One of the main challenges in finding the length of a curve is that it requires advanced mathematical techniques, such as integration, which can be difficult to grasp. Additionally, the length of a curve may also be infinite in some cases, making it impossible to find a definite value.

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