calculate-D-x-y-1-x-2-y-2-dxdy-with-D-x-y-R-2-x-0-y-0-x-2-y-2-lt-1-

Question Number 46846 by maxmathsup by imad last updated on 01/Nov/18

c a l c u l a t e \int \int_{D} \frac{x + y}{\sqrt{1 - x^{2} - y^{2}}} d x d y w i t h D = {(x, y) \in R^{2} / x ⩾ 0, y ⩾ 0, x^{2} + y^{2} < 1}

Commented by maxmathsup by imad last updated on 03/Nov/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 e (r, θ) \to φ (r, θ) = (x, y) w i t h

x = r c o s θ a n d y = r s i n θ \Rightarrow

\int \int_{D} f (x, y) d x d y = \int \int_{w} f o φ (r, θ) r d r d θ = \int \int_{0 ⩽ r ⩽ 1 a n d 0 ⩽ θ ⩽ \frac{π}{2}} \frac{r c o s θ + r s i n θ}{\sqrt{1 - r^{2}}} r d r d θ

= \int_{0}^{1} \frac{r^{2}}{\sqrt{1 - r^{2}}} d r \int_{0}^{\frac{π}{2}} (c o s θ + s i n θ) d θ b u t \int_{0}^{\frac{π}{2}} (c o s θ + s i n θ) d θ

= {[s i n θ - c o s θ]}_{0}^{\frac{π}{2}} = 1 + 1 = 2 a l s o c h a n g e m e n t r = s i n t g i v e

\int_{0}^{1} \frac{r^{2}}{\sqrt{1 - r^{2}}} d r = \int_{0}^{\frac{π}{2}} \frac{{s i n}^{2} t}{c o s t} c o s t d t = \int_{0}^{\frac{π}{2}} \frac{1 - c o s (2 t)}{2} d t

= \frac{π}{4} - \frac{1}{4} {[s i n (2 t)]}_{0}^{\frac{π}{2}} = \frac{π}{4} \Rightarrow \int \int_{D} \frac{x + y}{\sqrt{1 - x^{2} - y^{2}}} d x d y = \frac{π}{2} .