let-D-x-y-R-2-x-2-y-2-lt-1-find-the-value-of-D-dxdy-x-2-y-2-2-

Question Number 36192 by prof Abdo imad last updated on 30/May/18

l e t D = {(x, y) \in R^{2} / x^{2} + y^{2} < 1}

f i n d t h e v a l u e o f \int \int_{D} \frac{d x d y}{x^{2} + y^{2} + 2}

Commented by abdo imad last updated on 31/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) /

x = r c o s θ a n d y = r s i n θ w e h a v e

\int \int_{D} \frac{d x d y}{x^{2} + y^{2} + 2} = \int \int_{0 < r < 1 a n d 0 ⩽ θ ⩽ 2 π} \frac{r d r d θ}{r^{2} + 2}

= \int_{0}^{1} \frac{r d r}{r^{2} + 2} . \int_{0}^{2 π} d θ = 2 π {[\frac{1}{2} l n (r^{2} + 2)]}_{0}^{1}

= π {l n (3) - l n (2)} .