Stuck on arc length problem

In summary, the individual was given parametric equations and found the derivatives. They are stuck on the integral and are seeking help on how to simplify it. One approach suggested is to take the square root of the common factor of sin^t and integrate using substitution. Another approach is to take u=cos(t) and work with the integral of sqrt(1+4u^2) du.
  • #1
dark_omen
9
0
Okay, so I was given the parametric equations of x = (cos(t))^2 and y = cos(t). So I found dy/dt = -sin(t) and dx/dt = -2sin(t)cos(t). This is where I am getting stuck, so I have the L = integral from 0 to 4pi (sqrt((dx/dt)^2+(dy/dt)^2)) , but I don't know how to simplify this to get the answer to the problem. Can anyone help, thanks!
 
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  • #2
Take the square root of the common factor of sin^t, and then integrate either by substitution or by inspection.

Regards,
George
 
  • #3
Sorry, forget the "by inspection" part of my previous post.

You end up with

[tex]\int_{0}^{4\pi} \sqrt{1 + 4 \cos^2 t} \sin t dt,[/tex]

right?

I don't see a fast way to do this integral, but I can see how to do it using a couple of substitutions.

What might be a good first substitution?

Regards,
George
 
  • #4
Also, make sure to take the absolute value of the sin(t) you pull out of the square root.
 
  • #5
I took 1 + 4cos(x) as u, and I could integrate that, but when I solved for the length it was 0.
This is what I did:
- 1/4 * integral(sqrt(u^2)du)
- 1/4 * ((u)^2/(2))evaluated @ 0 to 4pi
 
  • #6
dark_omen said:
I took 1 + 4cos(x) as u, and I could integrate that, but when I solved for the length it was 0.
This is what I did:
- 1/4 * integral(sqrt(u^2)du)
- 1/4 * ((u)^2/(2))evaluated @ 0 to 4pi
If u = 1+ 4cos(t), then u^2 would be 1 + 8 cos(t) + 16 cos^2(t) !
So wrong u!

Now, you should take u= cos(t) and you will have something like the integral of sqrt(1+4 u^2)
 

1. What is an arc length problem?

An arc length problem is a mathematical problem that involves finding the length of a curved line or arc. This type of problem is commonly encountered in geometry and calculus.

2. How do you calculate the arc length?

The formula for calculating the arc length is L = rθ, where L is the arc length, r is the radius of the circle or curve, and θ is the angle subtended by the arc in radians. Alternatively, you can use the formula L = 2πr(n/360), where n is the central angle in degrees.

3. What are some real-world applications of arc length problems?

Arc length problems have many real-world applications, such as measuring the length of a curved road or track, determining the distance traveled by a satellite in orbit, and calculating the amount of fencing needed for a curved property boundary.

4. What are some common strategies for solving arc length problems?

One common strategy for solving arc length problems is to use the arc length formula mentioned above. Another approach is to break the curved line into smaller segments and use the Pythagorean theorem to find the length of each segment. In more complex cases, calculus techniques such as integration may be necessary.

5. How can I check if my answer to an arc length problem is correct?

If you are using a formula to solve an arc length problem, you can double-check your answer by plugging it back into the formula and making sure it gives you the original values. Additionally, you can use a calculator or computer program to verify your answer. Finally, if the problem is related to a real-world scenario, you can physically measure the arc length to see if it matches your calculated answer.

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