Arc length of vector function - the integral seems impossible

In summary: So in summary, you have an integral over ##\|(e^x \cdot F(x))'\|=\|e^x(F'(x)+F(x))\|=2e^x \sqrt{(\cos 2x -\sin 2x)^2+(\sin 2x+\cos 2x)^2+1^2}## where ##F(x)=(\cos 2x,\sin 2x,1).##
  • #1
overpen57mm
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Homework Statement
Find the arc length from 0-3pi for v(x)=(e^x cos(2x), e^x sin(2x), e^x)
Relevant Equations
Arc length formula for vector equations
The vector equation is ## v(x)=(e^x cos(2x), e^x sin(2x), e^x) ##

I know the arc-length formula is ## S=\int_a^b \|v(x)\| \,dx ##

I found the derivative from a previous question dealing with this same function, but the when I plug it into the arc-length function I get an integral that I've tried and tried but just can't get anywhere with. The complexity of the problem also makes me think that I might be approaching it from the wrong direction. Here is the integral as I understand it: $$ \int{\sqrt{ (e^{2x}) ( (-2\sin(2x) + \cos(2x))^2 + (2\cos(2x)+\sin(2x))^2 + 1 ) }} \, dx $$

I would appreciate any tips on the integral or the problem as a whole, if there's another way to solve it that I haven't seen.
 
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  • #2
You have an integral over ##\|(e^x \cdot F(x))'\|=\|e^x(F'(x)+F(x))\|=2e^x \sqrt{(\cos 2x -\sin 2x)^2+(\sin 2x+\cos 2x)^2+1^2}## where ##F(x)=(\cos 2x,\sin 2x,1).##
 
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  • #3
fresh_42 said:
You have an integral over ##\|(e^x \cdot F(x))'\|=\|e^x(F'(x)+F(x))\|=2e^x \sqrt{(\cos 2x -\sin 2x)^2+(\sin 2x+\cos 2x)^2+1^2}## where ##F(x)=(\cos 2x,\sin 2x,1).##
Thanks, I guess I was blinded by the derivative I got from the previous question.
 

FAQ: Arc length of vector function - the integral seems impossible

1. What is the arc length of a vector function?

The arc length of a vector function is the length of the curve traced out by the vector function. It can be calculated using the integral of the magnitude of the derivative of the vector function.

2. Why does the integral seem impossible to calculate for arc length of a vector function?

The integral for arc length of a vector function can seem impossible to calculate because it involves finding the magnitude of the derivative of the vector function, which can be a complex mathematical expression. Additionally, the integral may not have a closed-form solution and may need to be approximated numerically.

3. Is there an easier way to calculate the arc length of a vector function?

Yes, there are some special cases where the arc length of a vector function can be calculated using simpler methods. For example, if the vector function is a straight line, the arc length can be calculated using the distance formula.

4. Can the arc length of a vector function be negative?

No, the arc length of a vector function cannot be negative. It represents the distance traveled along the curve, so it will always be a positive value.

5. How is the arc length of a vector function used in real-world applications?

The arc length of a vector function is used in various fields such as physics, engineering, and computer graphics to calculate the distance traveled by an object or the length of a curve. It is also used in optimization problems to find the shortest path between two points.

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