Finding the length of a curve

In summary: ONLY when b is positiveOk I will try that, thanks for the help!In summary, the formula for finding the length of a curve is square root(1 + (dy/dx)^2 dx). To find the length of the curve 24xy = x^4 + 48 between (2, 4/3) and (3, 43/24), the integration was done by taking the derivative of 24xy = x^4 + 48 and squaring it, then adding 1 and taking the square root. From there, the common denominator was found and the property was used to integrate the equation. However, further steps are needed to find the final answer.
  • #1
rappa
8
0
Question: Find the length of the curve 24xy = x^4 + 48 between (2, 4/3) and (3, 43/24). Answer is supposed to be 9/8.

This is the formula (I think): square root(1 + (dy/dx)^2 dx)

I tried to integrate and I did it like this:

dy/dx = 3/24 x^2 - 2x^-2 or 3/24 x^2 - 2/x^2

then: square root(1 + (3/24 x^2 -2x^-2)^2)

= square root(1 + (1/64 x^4 - 4x^-4))

now integration:

(1 + (1/64 x^4 - 4x^-4))^(1/2)

= (2/3) (1 + (1/64 x^4 - 4x^-4))^(3/2) (x + (1/320 x^5 + 4/3 x^-3)

That's how far I got, I think it's wrong. If anyone could help me out, I would appreciate it. Thanks.
 
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  • #2
How did you come up with your y'
 
  • #3
24xy = x^4 + 48

y = (x^4 + 48) / 24x

y = 1/24 x^3 + 2/x or 2x^-1

dy/dx = 3/24 x^2 - 2x^-2
 
  • #4
Now show me how you do this:

then: square root(1 + (3/24 x^2 -2x^2)^2)

= square root(1 + (1/64 x^4 - 4x^-4))

[You probably mean 1 + (3/24 x^2 - 2/x^2)^2)]
 
  • #5
the 1+ part isn't squared, just the inside bracket. I just left the square root part, but I only squared the inside.
 
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  • #6
My post does not indicate that it is, does it?

My first point was that you wrote 2x^2 whereas you meant to write 2/x^2 or 2x^(-2)

My second point is show me how (3/24 x^2 - 2/x^2)^2 = 1/64 x^4 - 4x^(-4)
 
  • #7
is it: 1/64 x^4 - 1/2 + 4/x^4 ?
 
  • #8
That looks better
 
  • #9
ok so now I have:

(2/3) (1 + (1/64 x^4 - 1/2 + 4/x^4))^(3/2) (x + (1/320 x^5 - 1/2x - 4/3 x^-3)
 
  • #10
I have no idea how you got that, you have to show more steps, for example next step would be to show what happens when you add 1 to what you had and take sqrt
 
  • #11
We can't add the one, and we can't just take the square root either. Have to just integrate it straight from the square root and plug in the numbers to get the length.

This is what I did for the integration

(1/64 x^4 - 1/2 + 4/x^4)^(1/2)

= (2/3) (1 + (1/64 x^4 - 1/2 + 4/x^4))^(3/2) (x + (1/320 x^5 - 1/2x - 4/3 x^-3)
 
  • #12
...why not?

The steps are:

1) Take the derivative
2) Square it
3) add 1 to it
4) take square root of that
5) integrate
 
  • #13
Ok here goes, I hope that works...

(1 + (1/64 x^4 - 1/2 + 4/x^4)

65/64 x^4 + 1/2 + 5/x^4

so now square root of that:

root(65/65) x^2 + root (1/2) + root (5) / x^2


that looks bad, that's why I thought to do it the other way:

(1/64 x^4 - 1/2 + 4/x^4)^(1/2)

integrates to:

(2/3) (1 + (1/64 x^4 - 1/2 + 4/x^4))^(3/2) (x + (1/320 x^5 - 1/2x - 4/3 x^-3)
 
  • #14
You should know from algeba that sqrt(a+b) is NOT sqrt(a) + sqrt(b), stop thinking that.

To work with what you have, find the common denominator and then use the property that sqrt(a/b) DOES equal sqrt(a)/sqrt(b)
 

What is meant by "finding the length of a curve"?

Finding the length of a curve refers to calculating the total distance along a curved line. This can be useful in various fields such as mathematics, physics, and engineering.

Why is it important to find the length of a curve?

Knowing the length of a curve can help in understanding the behavior of a system, predicting future patterns, and making accurate measurements in real-world applications.

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

The length of a curve can be calculated using various methods such as integration, numerical approximations, or using specialized formulas for specific types of curves. The process involves breaking the curve into smaller segments and adding up their lengths to get an approximation of the total length.

Is it possible to find the exact length of a curve?

In most cases, it is not possible to find the exact length of a curve due to its infinite number of points. However, with more precise methods and advanced mathematical techniques, the length can be calculated to a higher degree of accuracy.

What are some real-world applications of finding the length of a curve?

Finding the length of a curve is used in various fields such as calculating the distance traveled by a moving object, designing roller coasters and other curved structures, predicting the path of a projectile, and measuring the perimeter of irregular shapes.

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