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camillesmith6630
25.12.2021 •
Mathematics
Use Green's Theorem to evaluate
integrate C (F · dr). (Check the orientation of the curve before applying the theorem.)
F(x, y) = vector ( sqrt(x)+4y^3, 4x^2 + sqrt(y) )
C consists of the arc of the curve y = sin(x) from (0, 0) to (pi, 0) and the line segment from (pi, 0) to (0, 0)
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Ответ:
By Green's theorem, the line integral of F along C is equal to the integral of the curl of F (two-dimensional curl, that is) over the region bounded by C, where C is a generic path that is oriented counterclockwise. However, our C run clockwise, so we multiply the following by -1.
where R is the set
Compute the double integral:
Integrating with respect to y is trivial:
Integrating by parts with
u = x ⇒ du = dx
dv = sin(x) dx ⇒ v = -cos(x)
gives
while in the other integral, we have by substitution
Then our last integral evaluates to
Ответ:
It’s the bottom answer.
Step-by-step explanation:
hope I helped