Activity 13.4.4.
Calculate each of the following line integrals.
(a)
\(\int_C \vF\cdot d\vr\) if \(\vF(x,y) = \langle 2x,2y\rangle\) and \(C\) is the line segment from \((1,2)\) to \((-1,0)\text{.}\)
(b)
\(\int_C \vG\cdot d\vr\) if \(\vG(x,y) = \langle 4x^3-12y\cos(xy),9y^2-12x\cos(xy)\rangle\) and \(C\) is the portion of the unit circle from \((0,-1)\) to \((0,1)\text{.}\)
(c)
\(\int_C \vH\cdot d\vr\) if \(\vH(x,y,z) = \langle H_1,H_2,H_3\rangle\) with
\begin{align*}
H_1(x,y,z) \amp = e^{z^2}+2xy^3z+\cos(x)-y^3\sin(x)\\
H_2(x,y,z) \amp = 2ye^{y^2}+3x^2y^2z+3y^2z^2+3y^2\cos(x)\\
H_3(x,y,z) \amp = x^2y^3+2xze^{z^2}+2y^3z-4z^3
\end{align*}
and \(C\) is the curve consisting of the line segment from \((1,1,1)\) to \((3,0,3)\text{,}\) followed by the line segment from \((3,0,3)\) to \((1,5,-1)\text{,}\) followed by the line segment from \((1,5,-1)\) to \((0,0,0)\text{.}\)
