Using Symmetry in Double Integrals
2
Using symmetry to simplify the calculation of the double integral Let D be a bounded region ( in R 2 ) 1) If D is symmetric about the y-axis and E = D ∩ { (x,y) : x ≥ 0 } = The right halve of D . Then: a. ∬ D f(x,y) dx dy = 0 ; if f is odd in x. b. ∬ D f(x,y) dx dy = 2 ∬ E f(x,y) dx dy ; if f is even in x. Example(1) -6 6 0 3 ( y sinx + 5x 3 y 2 )dy dx = 0 Example(2) -6 6 0 3 ( y cosx + 5x 4 y 2 )dy dx = 2 0 6 0 3 ( 2y cosx + x 4 y 2 )dy dx 2) If D is symmetric about the x-axis and E = D ∩ { (x,y) : y ≥ 0 } = The upper halve of D. Then :
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Using Symmetry in Double Integrals
Transcript of Using Symmetry in Double Integrals
Using symmetry to simplify the calculation of the double integral
Using symmetry to simplify the calculation of the double integralLet D be a bounded region ( in R2 )
1) If D is symmetric about the y-axis and E = D { (x,y) : x 0 } = The right halve of D . Then:
a. D f(x,y) dx dy = 0 ; if f is odd in x.b. D f(x,y) dx dy = 2 E f(x,y) dx dy ; if f is even in x.Example(1) -6( 6 0( 3 ( y sinx + 5x3 y2 )dy dx = 0
Example(2) -6( 6 0( 3 ( y cosx + 5x4 y2 )dy dx
= 2 0( 6 0( 3 ( 2y cosx + x4 y2 )dy dx 2) If D is symmetric about the x-axis and E = D { (x,y) : y 0 } = The upper halve of D. Then :
a. D f(x,y) dx dy = 0 ; if f is odd in y.b. D f(x,y) dx dy = 2 E f(x,y) dx dy ; if f is even in y.
Example(1) 0( 3 -6( 6 ( x siny + 5y3 x2 )dy dx = 0
Example(2) 0( 3 -6( 6 ( x cosy + 5y4 x2 )dy dx
= 2 0( 3 0( 6 ( x cosy + 5y4 x2 )dy dx
