Arc Length (Set up the Integral)

In summary, arc length is the distance along a curve, measured in units of length, from one point to another. It can be calculated by setting up an integral and integrating the square root of the sum of the squares of the derivatives of the curve's equation with respect to the variable of integration. The purpose of setting up an integral is to find the exact value of the arc length, rather than estimating it with a ruler or other measuring tool. The arc length of a curve cannot be negative, it is always a positive value. There are some special cases when setting up an integral for arc length, such as when dealing with a parametric curve or when splitting the integral into multiple intervals for more complex curves.
  • #1
johnhuntsman
76
0
x = t + cot t
y = t - sin t
0 ≤ t ≤ 2π

Somehow the answer is:

2π
∫sqrt(3 - 2*sin t - 2*cos t) dt
0

I'm afraid I don't know where to start on this one. I don't need someone to walk me through it (probably) but a point in the right direction would be appreciated.
 
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  • #2
Nevermind this post everyone. I made a foolish mistake. I'm supposed to do sqrt[(x')^2 + (y')^2] for anyone who makes the same mistake and comes across this in a Google search. It was in an earlie rpart of the chapter I'm working on even. Sorry everyone.
 

1. What is the definition of arc length?

Arc length is the distance along a curve, measured in units of length, from one point to another. It is similar to measuring the length of a straight line, but takes into account the curvature of the curve.

2. How is the arc length of a curve calculated?

The arc length of a curve can be calculated by setting up an integral and integrating the square root of the sum of the squares of the derivatives of the curve's equation with respect to the variable of integration.

3. What is the purpose of setting up an integral to find arc length?

Setting up an integral allows us to find the exact value of the arc length, rather than estimating it with a ruler or other measuring tool. It also allows us to find the arc length of more complex curves that cannot be measured directly.

4. Can the arc length of a curve be negative?

No, the arc length of a curve cannot be negative. It is always a positive value, representing the distance along the curve.

5. Are there any special cases when setting up an integral for arc length?

Yes, there are a few special cases when setting up an integral for arc length. For example, when dealing with a parametric curve, the integral formula is slightly different. Also, for some curves, it may be necessary to split the integral into multiple intervals to accurately calculate the arc length.

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