# Thread: Line Integral Question: "Suppose that the length of the path C is L...."

1. ## Line Integral Question: "Suppose that the length of the path C is L...."

(a) Consider the line integralI =The integral of Fdr along the curve c.

i. Describe, using words and formulas, how you would go about evaluating the integral I.
ii. Suppose that the length of the path C is L. What is the value of I if the vector field F isnormal to C at every point of C?
iii. What is the value of I if the vector field F is is a unit vector pointing in the negativedirection along the curve at every point of C?

I have an understanding of how to figure out the first part of the problem, as I would multiply my vector field f by the derivative of a given r vector. However, I don't know how I would go about solving the second and third part since there are no defined vectors for C. Any help on this would be greatly appreciated.

2. iii. What is the value of I if the vector field F is is a unit vector pointing in the negative direction along the curve at every point of C?
First, in order to have a "negative direction", you have to have a "positive direction"! I interpret that to mean "in the direction of increasing parameter. In that case "pointing in the negative direction" means tangent to the curve in the opposite direction to that positive direction. In integrating a unit tangent vector "in the positive direction" would give the length of the curve, Integrating a unit tangent vector in the "negative direction" gives the negative of the length of C.