Exercise 17

For each of the following line integrals:

• Determine if the Fundamental Theorem of Calculus for Line Integrals applies.
• If the Fundamental Theorem of Calculus for Line Integrals applies, then find the potential function and use this to evaluate the line integral
• If the Fundamental Theorem of Calculus for Line Integrals does not apply, then describe where the process laid out in this activity fails.

1. The line integral of $\vF=\langle yz,xz,y\rangle$ along the helix of radius of 3 given by $\vr(t) =\langle 3\sin(t), 3\cos(t), \frac{4}{\pi} t\rangle$ as $-\pi\leq t\leq \pi$.

2. The line integral of $\vF=\langle \sin(yz), xz\cos(yz)-z\sin(y),xy\cos(yz)+\cos(y)\rangle$ along the line segment from $(0,\pi,3)$ to $(2,-1,2\pi)$.

3. The integral$\int_C y^2\, dx +2xy\, dy$ where $C$ is the parabolic path along $y=x^2$ from $(-1,1)$ to $(4,16)$.