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=-3y^2 + x^2 Using Green's theorem, the line integral of F along C is equal to the double integral of the curl of F over the region R: ∮Fdr=∬curlFdA Since C is traversed once in a counterclockwise direction, the line integral becomes: ∮Fdr=∬curlFdA=∬-3y^2+x^2dA To evaluate this double integral, we'll switch to polar coordinates where x=rcosθ and y=rsinθ, and dA=rdrdθ: ∮Fdr=∬-3(rsinθ)^2+(rcosθ)^2rdrdθ =∬-3r^3sin^2θ+r^3cos^2θdrdθ Now, we can evaluate the double integral by splitting it into two separate integrals: ∮Fdr=∫[0,2π]∫[0,a](-3r^3sin^2θ+r^3cos^2θ)drdθ Integrating the inner integral with respect to r: dθ Now, we use the trigonometric identities sin^2θ=(1 - cos2θ)/2 and cos^2θ=(1 + cos2θ)/2 to simplify:

Name: 3y2 x2 using greens theorem the line integral of f along c is equal to the double integral of the curl of f over the region r fdrcurlfda since c is traversed once in a counterclockwise dire 87383
Uploaded: 2023-07-21T07:25:27-08:00
Duration: 4 min 27 s
Channel: Adi S
Description: 3y2 x2 using greens theorem the line integral of f along c is equal to the double integral of the curl of f over the region r fdrcurlfda since c is traversed once in a counterclockwise dire 87383

          =-3y^2 + x^2
Using Green's theorem, the line integral of F along C is equal to the double integral of the curl of F over the region R:
∮Fdr=∬curlFdA
Since C is traversed once in a counterclockwise direction, the line integral becomes:
∮Fdr=∬curlFdA=∬-3y^2+x^2dA
To evaluate this double integral, we'll switch to polar coordinates where x=rcosθ and y=rsinθ, and dA=rdrdθ:
∮Fdr=∬-3(rsinθ)^2+(rcosθ)^2rdrdθ
=∬-3r^3sin^2θ+r^3cos^2θdrdθ
Now, we can evaluate the double integral by splitting it into two separate integrals:
∮Fdr=∫[0,2π]∫[0,a](-3r^3sin^2θ+r^3cos^2θ)drdθ
Integrating the inner integral with respect to r:
dθ
Now, we use the trigonometric identities sin^2θ=(1 - cos2θ)/2 and cos^2θ=(1 + cos2θ)/2 to simplify:

3y2 x2 using greens theorem the line integral of f along c is equal to the double integral of the curl of f over the region r fdrcurlfda since c is traversed once in a counterclockwise dire 87383

Added by Christina S.

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