calculate ∫∫_w x(√(x^2 +y^2 )) dxdy w ={(x,y)/ x^2 +y^2 ≤3 }


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Question Number 34716 by abdo mathsup 649 cc last updated on 10/May/18
$calculate ∫∫_w x(√(x^2 +y^2 )) dxdy w ={(x,y)/ x^2 +y^2 ≤3 }$
$c a l c u l a t e \int \int_{w} x \sqrt{x^{2} + y^{2}} d x d y$ $w = {(x, y) / x^{2} + y^{2} ⩽ 3}$
Commented by prof Abdo imad last updated on 11/May/18

$l e t c o n s i d e r t h e d i f f e o m o r p h i s m$ $(r, θ) \to (x, y) = (r c o s θ, r s i n θ) / 0 ⩽ r ⩽ \sqrt{3}$ $a n d - π ⩽ θ ⩽ π$ $I = \int \int_{0 ⩽ r ⩽ \sqrt{3} a n d - π ⩽ θ ⩽ π} r c o s θ r . r d r d θ$ $= \int_{0}^{\sqrt{3}} r^{3} d r . \int_{- π}^{π} c o s θ d θ = \frac{1}{4} {(\sqrt{3})}^{3} {[s i n θ]}_{- π}^{π} = 0$ $I = 0$
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