Troublesome Arc Length Problem

I suspect that you may have made an error in computation as you worked through the problem or that the function is not what the instructor gave you. In summary, the problem is to find the arc length of a given function, but it might be necessary to confirm with the instructor that the function is correct and to check for any errors in computation.
  • #1
defecritus
4
0
1.) The problem is:

Find the arc length of f(x)= x^3/3-1/(4x) from x=1 to 2

2.) Relevant formulas:

ds = √(1+(dy/dx))

abs(L) = ∫ds

3.) My work so far:

f'(x)= x^2+1/(4x^2)

abs(L) = ∫(from 1 to 2) √(1+(x^2+1/(4x^2))^2 dx
= ∫(from 1 to 2) √(1+(x^4+1/2+1/(16x^4)) dx
= ∫(from 1 to 2) √(x^4+3/2+1/(16x^4)) dx
= ∫(from 1 to 2) √((16x^8+24x^4+1)/(16x^4) dx
= 1/4∫(from 1 to 2) √((16x^8+24x^4+1)/(x^4) dx
= 1/4∫(from 1 to 2) √(16x^8+24x^4+1)/(x^2) dx

Generally, I would suspect the numerator would be able to be factored and that would make this integral much easier to solve. I'm stumped after this part, because it seems that the numerator is irreducible.

I'm up for any help, I'm not asking for the answer. I really just need guidance from this point...that includes the possibility that the professor may have made a typo. I really wanted to consider that I may just not be aware of the next step before accusing the professor of fat fingering this. Thanks for the help.
 
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  • #2
defecritus said:
1.) The problem is:

Find the arc length of f(x)= x^3/3-1/(4x) from x=1 to 2
Are you sure this is the right function? Things would work out if the above was a sum, not a difference.
defecritus said:
2.) Relevant formulas:

ds = √(1+(dy/dx))
Your formula is neither relevant nor correct. Fortunately you used the right formula in your work.

##ds = \sqrt{1 + (dy/dx)^2}dx##
defecritus said:
abs(L) = ∫ds

3.) My work so far:

f'(x)= x^2+1/(4x^2)

abs(L) = ∫(from 1 to 2) √(1+(x^2+1/(4x^2))^2 dx
= ∫(from 1 to 2) √(1+(x^4+1/2+1/(16x^4)) dx
= ∫(from 1 to 2) √(x^4+3/2+1/(16x^4)) dx
= ∫(from 1 to 2) √((16x^8+24x^4+1)/(16x^4) dx
= 1/4∫(from 1 to 2) √((16x^8+24x^4+1)/(x^4) dx
= 1/4∫(from 1 to 2) √(16x^8+24x^4+1)/(x^2) dx

Generally, I would suspect the numerator would be able to be factored and that would make this integral much easier to solve. I'm stumped after this part, because it seems that the numerator is irreducible.

I'm up for any help, I'm not asking for the answer. I really just need guidance from this point...that includes the possibility that the professor may have made a typo. I really wanted to consider that I may just not be aware of the next step before accusing the professor of fat fingering this. Thanks for the help.
I would contact the instructor to confirm that the function you wrote is what he intended. Because of the radical in the integral, arc length problems are generally to difficult to integrate using the techniques that are taught in beginning calculus courses. The functions have to be "cooked up" so that 1 + (f'(x))2 is a perfect square, so that its square root can be simplified.
 

1. How do you solve a troublesome arc length problem?

Solving a troublesome arc length problem involves using the formula L=2πrθ/360°, where L is the length of the arc, r is the radius of the circle, and θ is the central angle in degrees. Plug in the given values and solve for L.

2. What is the difference between arc length and arc measure?

Arc length refers to the actual length of an arc on a circle, while arc measure refers to the angle subtended by the arc at the center of the circle. They are related by the formula L=2πrθ/360°, where L is the arc length, r is the radius, and θ is the arc measure in degrees.

3. When do I use radians instead of degrees in arc length problems?

Radians are used when the arc measure is given in radians instead of degrees. To convert from degrees to radians, use the formula θ (in radians) = θ (in degrees) * π/180°. Once the arc measure is in radians, you can use the formula L=rθ to find the arc length.

4. Can you use the arc length formula for a sector of a circle?

Yes, the arc length formula L=2πrθ/360° can be used for any arc on a circle, including a sector. The only difference is that the central angle θ will be the angle of the sector, not the entire circle.

5. How do I find the arc length if the radius is not given?

If the radius is not given, but the diameter is known, you can use the formula L=πdθ/360°, where L is the arc length, d is the diameter, and θ is the central angle in degrees. If neither the radius nor diameter is given, you will need to use other given information, such as the arc measure or the length of the chord, to find the radius before using the arc length formula.

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