Activity 13.5.4

Calculate each of the following line integrals.

1. $\int_C \vF\cdot d\vr$ if $\vF(x,y) = \langle 2x,2y\rangle$ and $C$ is the line segment from $(1,2)$ to $(-1,0)$.

2. $\int_C \vG\cdot d\vr$ 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 $(0,1)$.

3. $\int_C \vH\cdot d\vr$ if $\vH(x,y,z) = \langle H_1,H_2,H_3\rangle$ with $$\begin{aligned} H_1(x,y,z) & = e^{z^2}+2xy^3z+\cos(x)-y^3\sin(x) \\ H_2(x,y,z) & = 2ye^{y^2}+3x^2y^2z+3y^2z^2+3y^2\cos(x) \\ H_3(x,y,z) & = x^2y^3+2xze^{z^2}+2y^3z-4z^3 \end{aligned}$$ and $C$ is the curve consisting of the line segment from $(1,1,1)$ to $(3,0,3)$, followed by the line segment from $(3,0,3)$ to $(1,5,-1)$, followed by the line segment from $(1,5,-1)$ to $(0,0,0)$.