\(\int_C \nabla f\cdot d\vr\) 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 \((1,0)\text{.}\)
\(\int_C \nabla g\cdot d\vr\) 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 \((0,5,9\pi/2)\text{.}\)
\(\int_C \nabla h\cdot d\vr\) 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 \((1,1,1)\text{,}\) followed by the line segment from \((1,1,1)\) to \((-1,3,-2)\text{,}\) followed by the line segment from \((-1,3,-2)\) to \((0,0,10)\text{.}\)
