Question

Calc 3 homework please help due tonight!!!

Answer 1

Have you attempted anything yet?

Answer 2

And I get 12 pi. Try that and if that works I can explain it to you. Or more of, walk you through it.

Answer 3

an explanation would be helpful to me :)

Answer 4

Okay :)

Answer 5

The easiest one would be to notice that:
\[d \vec{a}=d \phi d z \hat{s}\]
If we write the vector field out in polar we get:
\[(x,y,z) \rightarrow = x \hat{x}+y \hat{y}+z \hat{z} = \cos(\phi) \hat{x}+\sin(\phi) \hat{y}+z \hat{z}=\hat{s}+z \hat{z} \rightarrow (1,0,z)\]
So we have:
\[\int\limits \vec{F} \cdot d \vec{a}=\int\limits_0^{2 \pi} \int\limits_{1-\cos(\phi)-\sin(\phi)}^{7-\cos(\phi)-\sin(\phi)}dz d \phi=\int\limits_0^{2 \pi} (7-1)d \phi = 6 \int\limits_0^{2 \pi}d \phi = 12 \pi\]

Answer 6

Since the cosines and sines cancel nicely :)

Answer 7

And all of this holds for x^2+y^2=1 so r=1 for all the change of variables.