Using integral to find length of curve

In summary, the problem is to determine which of two curves is shorter. The first curve is unknown and the second curve should be easier once the first one is found. The attempt at a solution involved using a formula and a trig substitution, but the integral of sec(x) is typically solved using a trick involving multiplying by (sec(x) + tan(x)) and a u-substitution.
  • #1
a1ccook
2
0

Homework Statement


Ok, the problem is there are two curves and I need to find out which is shorter. I can't find the first one so I'll just post that. The other should be easy after I learn how to get this one.

Homework Equations


The first curve is:
EQ.jpg


The Attempt at a Solution



I used the formula
EQ2.jpg
and worked it down to
EQ3.jpg
.

I've also seen some similar problems to this where people recommended trig substitutions. I tried substituting tan u in which gave me sec^2 u under the radical. If I go that route, I don't know the integral of sec or how you could carry on. The other way I thought you could do this was to just simplify (from the last picture I posted) and try to integrate that... but that would be rough. Any tips?
 
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  • #2
The integral of sec(x) is usually done with the trick of multiplying the numerator and denominator by (sec(x) + tan(x)) followed by a u-substitution. Try it.
 

1. What is the purpose of using integrals to find the length of a curve?

Integrals are mathematical tools used to calculate the area under a curve. By using integrals, we can also find the length of a curve by approximating it with small straight lines and summing up their lengths. This allows us to accurately calculate the length of a curve, which is useful in many fields of science and engineering.

2. How do you set up the integral to find the length of a curve?

To set up the integral, we first need to divide the curve into small segments. Then, we use the distance formula to find the length of each segment. Finally, we use the integral to sum up all the lengths of the segments, giving us the total length of the curve.

3. Can integrals be used to find the length of any type of curve?

Yes, integrals can be used to find the length of any type of curve, as long as the curve can be divided into small segments and the length of each segment can be calculated.

4. Are there any limitations to using integrals to find the length of a curve?

One limitation of using integrals is that it can be a time-consuming process, especially for complex curves. Additionally, if the curve has sharp turns or changes rapidly, the accuracy of the calculated length may be affected.

5. How is finding the length of a curve using integrals useful in real-world applications?

Finding the length of a curve using integrals is useful in various real-world applications, such as calculating the distance traveled by a moving object, designing curved structures in engineering, and determining the perimeter of irregularly shaped objects. It is also essential in fields such as physics, where the length of a curve is needed to calculate work and energy.

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