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Examples8.1

# Examples8.1 - sin x 1 x 2 ² dx ± x e y 1 y 2 ² dy...

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1 Example 1. Let C be the perimeter of the rectangle with sides x = 1 , y = 2 , x = 3 , and y = 3 . Evaluate the integral Z C (3 x 4 + 5) dx + ( y 5 + 3 y 2 - 1) dy Solution. Use Green’s theorem: Z C (3 x 4 + 5) dx + ( y 5 + 3 y 2 - 1) dy = Z 3 2 Z 3 1 ∂x ( y 5 + 3 y 2 - 1) - ∂y (3 x 4 + 5) dx dy = 0 .

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2 Example 2. Let F ( x, y ) = (2 y + e x ) i + ( x + sin y 2 ) j and C be the circle x 2 + y 2 = 1. Evaluate Z C F · d s . Solution. Let D be the unit disk bounded by C , then by Green’s theorem, Z C F · d s = ZZ D ∂x ( x + sin y 2 ) - ∂y (2 y + e x ) dx dy = ZZ D (1 - 2) dx dy = - π.
3 Example 3. Let C be the boundary of the rectangle with sides x = 1
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Unformatted text preview: + sin x 1 + x 2 ² dx + ± x + e y 1 + y 2 ² dy. Solution. Let R be the rectangle bounded by C . Then by Green’s theorem, Z C 2 y + sin x 1 + x 2 dx + x + e y 1 + y 2 dy = ZZ R ³ ∂ ∂x ± x + e y 1 + y 2 ²-∂ ∂y ± 2 y + sin x 1 + x 2 ²´ dxdy = Z 3 2 Z 3 1 ± 1 1 + y 2-2 1 + x 2 ² dxdy = (2 arctan y ) µ µ µ µ 3 2-(2 arctan x ) µ µ µ µ 3 1 = π 2-2 arctan 2 ....
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Examples8.1 - sin x 1 x 2 ² dx ± x e y 1 y 2 ² dy...

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