A use greens theorem to compute the area enclosed by

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Unformatted text preview: + 2 ) = 2+ 2≤ 2 2 θ 0 0 · = π4 2 E Let C be the closed curve given by parametric equations 3 sin3 for 0 ≤ ≤ 2π . (a) Use Green’s Theorem to compute the area enclosed by the curve C . 5 :) = 4 cos , = (b) Use Green’s Theorem to compute 2 (3 + 4 cos ) 2 + (3 +2 + ) C S (a) Let D be the region enclosed by the curve C . By Green’s theorem, we have − = = − (− ) C 1 = Area of D D D Thus, it suffices to evaluate 2π − = C 2π 3 −3 sin 0 2π 0 2 1 − cos 2 2 =12 0 2π 1 − 2 cos 2 + cos2 (2 ) =3 0 2π 2π 2 =6π + 3 sin4 4 cos = 12 cos (2 ) = 6π + 3 0 0 1 + cos(4 ) 2 = 6π + 3π = 9π (b) By Green’s theorem, we have (3 2 + 4 cos ) 2 + (3 +2 + ) C = ((3 2 +2 + 2 ) − (3 + 4 cos ) ) =2 D 1 = 18π :) D by part (a). E Find a positively oriented simple cl...
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This document was uploaded on 02/10/2014.

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