The discussion centers on calculating the line integral \(\oint _c xdy - ydx\) along a straight line segment from \((x_1, y_1)\) to \((x_2, y_2)\). While initially approached using Green's Theorem, it was concluded that the theorem is not applicable since the problem does not enclose an area. The correct approach is to evaluate the line integral directly, resulting in the expression \(2[(y_2 - y_1)(x_2 - x_1)]\).
PREREQUISITESStudents studying calculus, particularly those focusing on vector calculus and line integrals, as well as educators looking for examples of applying Green's Theorem correctly.
gtfitzpatrick said:Homework Statement
Calculate \oint _c xdy-ydx where C is the straight line segment from(x_1 , y_1) to (x_2 , y_2)
Homework Equations
The Attempt at a Solution
so from Greens theorem I get \oint _c xdy-ydx = \int\int \frac{\partial x}{\partial x} + \frac{\partial y}{\partial y} dxdy
=2 \int^{x_2}_{x_1}\int^{y_2}_{y_1} dydx
=2[(y_2 - y_1)(x_2 - x_1) ]