Line Integrals: Evaluating with Respect to Arc Length

In summary, the conversation discusses evaluating a line integral with respect to arc length. The integral is over the arc of the unit circle from (1,0) to (-1,0) traversed counterclockwise. The integral is parameterized using x = cos theta and y = sin theta, and the resulting integral is evaluated with the limits of theta going from 0 to pi. The final result is determined to be 0 due to the symmetry of the function with respect to the y-axis.
  • #1
wubie
Hello,

I must be having some sort of brain malfunction or something. First here is my question:

Evaluate the line integral with respect to arc length

The integral sub C of x*e^y ds where C is the arc of the unit circle from (1,0) to (-1,0) traversed counterclockwise.

Now if the circle is traversed counterclockwise then

-1 <= x <= 1 and 0 <= y <= 1.

I can parameterize the equation by noticing that

x = cos theta and y = sin theta.

Therefore the integral becomes

The integral of cos theta * e^sin theta ds

where ds is

( (-sin theta)^2 + (cos theta)^2 )^1/2 = 1

So the integral becomes

The integral of cos theta * e^sin theta dtheta.

Since the unit circle is traversed counterclockwise the angle goes from 0 to pi.

If I let sin theta = u then du = cos theta and the limits become zero and zero.

So if I integrate the integral of e^u du then I get

e^u with the limits of 0 and 0 which ends up to be zero.

This cannot be right can it? I am confused. I think I did everything correctly. Where am I going wrong?

Perhaps I am right?

Any help is appreciated. Thankyou.
 
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  • #2
Looks right to me, but then I've always sucked at path integrals, so. Maybe Matt will grace you with his presence.

It actually kinda seems right, too. It's an odd function with respect to the y-axis, and since we're going from -1 to 1, we have a symmetric domain. That suggests an answer of 0.

cookiemonster
 
  • #3
Yes, that's exactly right. "xey" is an odd function with respect to x and since the path of integration is symmetric with respect to the y-axis the integral is 0.

If it makes you feel better, you might try looking at exactly the same problem from (0,-1) to (0,1) instead. You get exactly the same integral except that theta now runs from -pi/2 to pi/2. Since you are making the substitution u= sin(theta), you integration will be from
-1 to 1 giving a non-zero result. That is, of course, correct since the function is not symmetric with respect to the x-axis.
 

1. What is a line integral?

A line integral is a mathematical concept that measures the total value of a function along a curve or line. It is used to calculate quantities such as work, mass, and energy in physics and engineering.

2. How is a line integral evaluated with respect to arc length?

When evaluating a line integral with respect to arc length, the curve is divided into small segments, and the value of the function at each point is multiplied by the length of the segment. These values are then summed up to find the total value of the integral.

3. What is the significance of evaluating a line integral with respect to arc length?

Evaluating a line integral with respect to arc length allows for the calculation of quantities that are independent of the parametrization of the curve. This makes it a more useful and versatile tool in various applications.

4. What are some applications of line integrals with respect to arc length?

Line integrals with respect to arc length are commonly used in physics and engineering to calculate work done by a force, mass of a wire, or energy stored in an object. They are also used in geometry to find the length of a curve.

5. How is the direction of integration determined in a line integral with respect to arc length?

The direction of integration is determined by the orientation of the curve. If the curve is given in a specific direction, then the line integral is evaluated in the same direction. If the curve is given in both directions, then the integral is split into two parts, and each is evaluated in the corresponding direction.

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