Chapter 16.3, Problem 32E

(a)

To determine

Find the value of the integral ${\displaystyle \underset{C}{\int }2x\cos ydx-x^2\sin ydy}$ along the parabola $y={\left(x-1\right)}^2$ from $\left(1,0\right)$ to $\left(0,1\right)$.

(b)

To determine

Find the value of the integral ${\displaystyle \underset{C}{\int }2x\cos ydx-x^2\sin ydy}$ along the line segment from $\left(-1,\pi \right)$ to $\left(1,0\right)$.

(c)

To determine

Find the value of the integral ${\displaystyle \underset{C}{\int }2x\cos ydx-x^2\sin ydy}$ along the $x\text{-axis}$ from $\left(-1,0\right)$ to $\left(1,0\right)$.

(d)

To determine

Find the value of the integral ${\displaystyle \underset{C}{\int }2x\cos ydx-x^2\sin ydy}$ along the asteroid $r\left(t\right)=\left({\cos }^{3}t\right)i+\left({\sin }^{3}t\right)j,0\le t\le 2\pi$ counter clock wise from $\left(1,0\right)$ back to $\left(1,0\right)$.