Activity 13.5.2

Calculate each of the following line integrals.

1. $\int_C \nabla f\cdot d\vr$ if $f(x,y) = 3xy^2 - \sin(x) + e^y$ and $C$ is the top half of the unit circle oriented from $(-1,0)$ to $(1,0)$.

2. $\int_C \nabla g\cdot d\vr$ if $g(x,y,z) = xz^2 - 5y^3\cos(z) + 6$ and $C$ is the portion of the helix $\vr(t) = \langle 5\cos(t),5\sin(t),3t\rangle$ from $(5,0,0)$ to $(0,5,9\pi/2)$.

3. $\int_C \nabla h\cdot d\vr$ if $h(x,y,z) = 3y^2e^{y^3} - 5x\sin(x^3z) + z^2$ and $C$ is the curve consisting of the line segment from $(0,0,0)$ to $(1,1,1)$, followed by the line segment from $(1,1,1)$ to $(-1,3,-2)$, followed by the line segment from $(-1,3,-2)$ to $(0,0,10)$.