Question

Here is another problem. There is a cute shortcut and a longer way to do it.

Answer 1

E. 1312

Make the path closed by adding x = 8, -8 <= y <= 0, and use Green's theorem. Then subtract the line integral that you added.

Answer 2

@sourwing , you are right. I will not publish the details for now. Let us see if some people can fill in the details.

Answer 3

join B->A path,

\(\large \mathbb{ \int_C \vec{F} . dr = \oint \vec{F} . dr - \int_{BA} \vec{F} . dr} \)

\(\large \mathbb{= \iint \limits_R curl \vec{F} ~ dA - \int_{BA} \vec{F} . dr} \)

\(\large \mathbb{= \iint \limits_R 7 ~ dA - \int_{-8}^0 4 \times 8 dt} \)

\(\large \mathbb{= 7 (16^2 - 8 \times 8/2) - 32 (8)} \)

\(\large \mathbb{=1312} \)