By Stokes' theorem, the line integral of $\vec F$ over $C$ is given by the surface integral of the curl of $\vec F$ over $S$ , where $S$ is the region of intersection of the plane $z=y+6$ and the cylinder $x^2+y^2=1$ with $S$ having positive/upward orientation.

Parameterize $S$ by

$\vec s(u,v)=u\cos v\,\vec\imath+u\sin v\,\vec\jmath+(u\sin v+6)\,\vec k$

with $0\le u\le1$ and $0\le v\le2\pi$ .

Take the normal vector to $S$ to be

$\vec s_u\times\vec s_v=-u\,\vec\jmath+u\,\vec k$

The curl of $\vec F$ is

$\nabla\times\vec F(x,y,z)=(1-x)\,\vec\imath-\vec\jmath+(z-2)\,\vec k$

Then the line integral is equivalent to

$\displaystyle\int_C\vec F\cdot\mathrm d\vec r=\iint_S(\nabla\times\vec F)\cdot\mathrm d\vec S$

$=\displaystyle\int_0^{2\pi}\int_0^1\bigg((1-u\cos v)\,\vec\imath-\vec\jmath+(u\sin v+4)\,\vec k\bigg)\cdot\bigg(-u\,\vec\jmath+u\,\vec k\bigg)\,\mathrm du\,\mathrm dv$

$=\displaystyle\int_0^{2\pi}\int_0^1(5u+u^2\sin v)\,\mathrm du\,\mathrm dv=\boxed{5\pi}$

4 0

3 years ago

Evaluate-32 + (2 – 6)(10)<br> The answer is -49

Drupady [299]

Ok thank you I think you answered in the wrong place

8 0

4 years ago

Truth or false every fraction has a limited number of equivalent frections .

xeze [42]

Answer:

False.

Since the number of numbers in the world is infinite, there can be an infinite number of equivalent fractions.

4 0

3 years ago

Read 2 more answers

Someone want to help me with some Geometry?

Aleks04 [339]

The second answer that uses = is not correct, because the pentagons are similar not congruent.
The third answer is not correct, because that sign means estimated, not similar.
I've never seen the fourth sign, so by process of elimination I believe the answer is the first one.

<span>ABCDE~QRSTU</span>

8 0

3 years ago