cslculate-0-1-2-x-y-e-x-y-dxdy-

Question Number 37279 by abdo.msup.com last updated on 11/Jun/18

c s l c u l a t e \int \int_{{[0, 1]}^{2}} (x - y) e^{- x - y} d x d y .

Commented by prof Abdo imad last updated on 18/Jun/18

l e t u s e t h e c h a n g e m e n t

(u, v) \to (x, y) = (x - y, x + y) = (φ_{1} (u, v), φ_{2} (u, v))

w e h a v e 0 ⩽ x ⩽ 1 a n d 0 ⩽ y ⩽ 1 \Rightarrow

- 1 ⩽ - y ⩽ 0 \Rightarrow - 1 ⩽ x - y ⩽ 1 a n d

0 ⩽ x + y ⩽ 2 \Rightarrow - 1 ⩽ u ⩽ 1 a n d 0 ⩽ v ⩽ 2 a n d

M_{j} = (\begin{matrix} \frac{\partial φ_{1}}{\partial u} \frac{\partial φ_{1}}{\partial v} \\ \frac{\partial φ_{2}}{\partial u} \frac{\partial φ_{2}}{\partial v} \end{matrix})

= (\begin{matrix}  \end{matrix})

w e h a v e x - y = u a n d x + y = v \Rightarrow 2 x = u + v

a n d 2 y = - u + v \Rightarrow x = \frac{1}{2} u + \frac{1}{2} v a n d

y = - \frac{1}{2} u + \frac{1}{2} v \Rightarrow

M_{j} = (\begin{matrix} \frac{1}{2} \frac{1}{2} \\ - \frac{1}{2} \frac{1}{2} \end{matrix}) a n d d e t (M_{j}) = \frac{1}{2}

\int \int_{{[0, 1]}^{2}} (x - y) e^{- (x + y)} d x d y

= \int \int_{- 1 ⩽ u ⩽ 1 a n d 0 ⩽ v ⩽ 2} u e^{- v} \frac{1}{2} d u d v

= \frac{1}{2} \int_{- 1}^{1} u d u . \int_{0}^{2} e^{- v} d v

= \frac{1}{2} {[\frac{u^{2}}{2}]}_{- 1}^{1} . {[- e^{- v}]}_{0}^{2} = 0

Answered by ajfour last updated on 11/Jun/18

A = \int_{0}^{1} \frac{1}{e^{y}} [\int_{0}^{1} {x e}^{- x} d x - y \int_{0}^{1} e^{- x} d x] d y

= \int_{9}^{1} \frac{1}{e^{y}} [- (1 + x) e^{- x} + {y e}^{- x}] ∣_{0}^{1} d y

= \int_{0}^{1} \frac{1}{e^{y}} [(\frac{y}{e} - \frac{2}{e}) - (y - 1)] d y

= \int_{0}^{1} [(\frac{1}{e} - 1) y + (1 - \frac{2}{e})] e^{- y} d y

= {- [(\frac{1}{e} - 1) y + (1 - \frac{2}{e})] e^{- y}

- (\frac{1}{e} - 1) e^{- y}} ∣_{0}^{1}

= - \frac{1}{e} [\frac{1}{e} - 1 + 1 - \frac{2}{e}] + 1 - \frac{2}{e}

- \frac{1}{e} (\frac{1}{e} - 1) + \frac{1}{e} - 1

= 0 .

Commented by prof Abdo imad last updated on 18/Jun/18

y o u r a n s w e r i s c o r r e c t s i r A j f o u r t h a n k s .

Commented by ajfour last updated on 18/Jun/18

t h a n k s f o r c o n f i r m i n g m y a n s w e r

S i r; m a k e s m e h a p p y .

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