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

Question Number 42395 by abdo.msup.com last updated on 24/Aug/18

c a l c u l a t e \int \int_{D} \frac{x y}{(1 + x^{2} + y^{2})} d x d y w i t h

D = {(x, y) \in R^{2} / 0 ⩽ x ⩽ 1, 0 ⩽ y ⩽ 1, x^{2} + y^{2} ⩽ 1}

Commented by maxmathsup by imad last updated on 25/Aug/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 (x, y) = (r c o s θ, r s i n θ)

\int \int_{D} f (x, y) d x d y = \int \int_{W} f o φ (t) ∣ J_{φ} ∣ d r d θ

= \int \int_{0 ⩽ θ ⩽ \frac{π}{2} a n d 0 ⩽ r ⩽ 1} \frac{r^{2} c o s θ s i n θ}{(1 + r^{2})} r d r d θ

= \int_{0}^{\frac{π}{2}} (\int_{0}^{1} \frac{r^{3}}{1 + r^{2}} d r) c o s θ s i n θ d θ = \int_{0}^{1} \frac{r^{3}}{1 + r^{2}} d r . (\frac{1}{2} \int_{0}^{\frac{π}{2}} s i n (2 θ) d θ) b u t

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

\int_{0}^{1} \frac{r^{3}}{1 + r^{2}} d r = \int_{0}^{\frac{π}{4}} \frac{{t a n}^{3} α}{(1 + {t a n}^{2} α)} (1 + {t a n}^{2} α) d α

= \int_{0}^{\frac{π}{4}} {t a n}^{3} α d α l e t A = \int_{0}^{\frac{π}{4}} {t a n}^{3} t d t

A = \int_{0}^{\frac{π}{4}} (1 + {t a n}^{2} t - 1) t a n t d t = \int_{0}^{\frac{π}{4}} (1 + {t a n}^{2} t) t a n t d t - \int_{0}^{\frac{π}{4}} t a n t d t

= {[t a n t t a n t]}_{0}^{\frac{π}{4}} - \int_{0}^{\frac{π}{4}} t a n t (1 + {t a n}^{2} t) d t - \int_{0}^{\frac{π}{4}} t a n t d t

= 1 - 2 \int_{0}^{\frac{π}{4}} t a n t d t - \int_{0}^{\frac{π}{4}} {t a n}^{3} t d t \Rightarrow 2 A = 1 - 2 \int_{0}^{\frac{π}{4}} t a n t d t \Rightarrow

A = \frac{1}{2} - \int_{0}^{\frac{π}{4}} \frac{s i n t}{c o s t} d t = \frac{1}{2} + {[l n ∣ c o s t ∣]}_{0}^{\frac{π}{4}} = \frac{1}{2} + l n (\frac{1}{\sqrt{2}}) = \frac{1}{2} - \frac{1}{2} l n (2) \Rightarrow

\int \int_{D} \frac{x y}{(1 + x^{2} + y^{2})} d x d y = \frac{l n (2)}{2} - \frac{1}{2} .

e r r o r a t t h e f i n a l l i n e \int \int \frac{x y}{1 + x^{2} + y^{2}} d x d y = \frac{1}{2} (\frac{l n (2)}{2} - \frac{1}{2})

= \frac{l n (2)}{4} - \frac{1}{4} .