The triangle is bounded by the lines y x 1 x 2 and y

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Unformatted text preview: x dy = cos x − x dy = 4. (a) To use Green’s Theorem the boundary curve must be oriented in the counterclockwise direction. Now ln(x2 + 1) dx − x dy = − ln(x2 + 1) dx − γ −γ Green s x dy = T heorem 0 −2 y 2 −4 1 4) dy = − y 3 − 4y 3 2 −2 y 2 x y2 4 R 2 2 − (−1) dx dy = − 8 −8 3 = −2 −2 (y 2 − x -4 32 =. 3 -2 (b) The region is a circular disk of radius 3 centered at (1, 3). Now Green s x dy = T heorem + − disk of radius 3 2 (1 − (−1)) dA = 2 γ (x − y ) dx + dA = 2 ( area of disk of radius 3 a disk of radius 3 ) = 2 (π 3 ) = 18π . 5. (a) We will use Green’s Theorem. The triangle is bounded by the lines y = x + 1, x = 2 and y = 1 and w...
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