let-a-gt-0-calculate-x-2-y-2-3-1-2-x-2-y-2-dxdy-

Question Number 34713 by abdo mathsup 649 cc last updated on 10/May/18

l e t a > 0 c a l c u l a t e \int \int_{x^{2} + y^{2} ⩽ 3} \frac{1}{2 + x^{2} + y^{2}} d x d y .

Commented by math khazana by abdo last updated on 11/May/18

l e t u d e ? t h e d i f f e o m o r p h i s m e x = r c o s θ,

y = r s i n θ w i t h 0 ⩽ r ⩽ \sqrt{3} a n d - π ⩽ θ ⩽ π

I = \int \int_{0 ⩽ r ⩽ \sqrt{3} a n d - π ⩽ θ ⩽ π} \frac{1}{2 + r^{2}} r d r d θ

= \int_{0}^{\sqrt{3}} \frac{r}{2 + r^{2}} d r . \int_{- π}^{π} d θ = 2 π \int_{0}^{\sqrt{3}} \frac{r}{2 + r^{2}} d r

= π {[l n (2 + r^{2})]}_{0}^{\sqrt{3}} = π {l n (5) - l n (2)} .