R is the region bounded by a circle of radius a centered at the origin and C is the boundary curve oriented counterclockwise. Φ(x,y)=y^2i+(4x+2xy)j Compute ∫ C Φ⋅dr by parametrizing C. Use Green's theorem to compute ∫ C Φ⋅dr without actually doing an integral. I need help parametrizing Φ
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∫c Φ ⋅ dr
= ∫c (y^2 dx + (4x + 2xy) dy)
= ∫∫R [(∂/∂x)(4x + 2xy) - (∂/∂y)(y^2)] dA, by Green's Theorem
= ∫∫R 4 dA
= 4 * (Area inside R, the circle of radius a)
= 4πa^2.
I hope this helps!
= ∫c (y^2 dx + (4x + 2xy) dy)
= ∫∫R [(∂/∂x)(4x + 2xy) - (∂/∂y)(y^2)] dA, by Green's Theorem
= ∫∫R 4 dA
= 4 * (Area inside R, the circle of radius a)
= 4πa^2.
I hope this helps!