find-1-x-2-y-2-4-and-y-0-dxdy-x-2-y-2-

Question Number 31090 by abdo imad last updated on 02/Mar/18

f i n d \int \int_{1 ⩽ x^{2} + y^{2} ⩽ 4 a n d y ⩾ 0} \frac{d x d y}{\sqrt{x^{2} + y^{2}}} .

Commented by abdo imad last updated on 06/Mar/18

l e t u s e t h e p o l a r c o o r d i n a t e s x = r c o s θ a n d y = r s i n θ

d u e t o d i f f e o m o r p h i s m e w e m u s t h a v e 1 ⩽ r ⩽ 2 a n d

0 ⩽ θ ⩽ π \Rightarrow I = \int_{0}^{π} (\int_{1}^{2} \frac{r d r}{\sqrt{r^{2}}}) d θ = \int_{0}^{π} (\int_{1}^{2} d r) d θ = \int_{0}^{π} d θ = π .