Perimeter of semicircular
2 answers:
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If
then
Differentiating with respec to gives
So is indeed conservative, and
Answer:
The result of the integral is 81π
Step-by-step explanation:
We can use Stoke's Theorem to evaluate the given integral, thus we can write first the theorem:
Finding the curl of F.
Given we have:
Working with the determinant we get
Working with the partial derivatives
Integrating using Stokes' Theorem
Now that we have the curl we can proceed integrating
where the normal to the circle is just since the normal is perpendicular to it, so we get
Only the z-component will not be 0 after that dot product we get
Since the circle is at z = 3 we can just write
Thus the integral represents the area of a circle, the given circle has a radius r = 3, so its area is
, so we get
Thus the result of the integral is 81π