Examples8.1

Examples8.1 - + sin x 1 + x 2 dx + x + e y 1 + y 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) ² dxdy = 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 ) ² dxdy = ZZ D (1 - 2) dxdy = - π.
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3 Example 3. Let C be the boundary of the rectangle with sides x = 1 ,y = 2 ,x = 3 and y = 3. Evaluate Z C ± 2 y
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Unformatted text preview: + sin x 1 + x 2 dx + x + e y 1 + y 2 dy. Solution. Let R be the rectangle bounded by C . Then by Greens theorem, Z C 2 y + sin x 1 + x 2 dx + x + e y 1 + y 2 dy = ZZ R x x + e y 1 + y 2 - y 2 y + sin x 1 + x 2 dxdy = Z 3 2 Z 3 1 1 1 + y 2-2 1 + x 2 dxdy = (2 arctan y ) 3 2-(2 arctan x ) 3 1 = 2-2 arctan 2 ....
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Examples8.1 - + sin x 1 + x 2 dx + x + e y 1 + y 2 dy....

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