Chapter 15.3, Problem 8ES

(a)

To determine

To prove: The line integral ${\displaystyle \underset{C}{\int }y\sin xdx-\cos xdy}$ over the curve $C$ is independent of the path.

(b)

To determine

To calculate: The line integral ${\displaystyle \underset{C}{\int }y\sin xdx-\cos xdy}$ over the curve $C$ which is a line segment from $\left(0,1\right)$ to $\left(\pi ,-1\right)$ .

(c)

To determine

To calculate: The integral ${\displaystyle \underset{\left(0,1\right)}{\overset{\left(\pi ,-1\right)}{\int }}y\sin xdx-\cos xdy}$ joins from $\left(0,1\right)$ to $\left(\pi ,-1\right)$ .