calculate-D-xy-e-x-2-y-2-dxdy-with-D-x-y-R-2-0-x-2-and-1-y-3-

Question Number 57320 by turbo msup by abdo last updated on 02/Apr/19

c a l c u l a t e \int \int_{D} x y e^{- x^{2} - y^{2}} d x d y

w i t h D = {(x, y) \in R^{2} / 0 ⩽ x ⩽ 2 a n d

1 ⩽ y ⩽ 3}

Commented by maxmathsup by imad last updated on 02/Apr/19

\int \int_{D} x y e^{- x^{2} - y^{2}} d x d y = \int_{0}^{2} x e^{- x^{2}} d x . \int_{1}^{3} y e^{- y^{2}} d y b u t

\int_{0}^{2} x e^{- x^{2}} d x = {[- \frac{1}{2} e^{- x^{2}}]}_{0}^{2} = - \frac{1}{2} {e^{- 4} - 1} = \frac{1}{2} {1 - e^{- 4})

\int_{1}^{3} y e^{- y^{2}} d y = {[- \frac{1}{2} e^{- y^{2}}]}_{1}^{3} = - \frac{1}{2} {e^{- 9} - e^{- 1}} = \frac{1}{2} {e^{- 1} - e^{- 9}} \Rightarrow

\int \int_{D} x y e^{- x^{2} - y^{2}} d x d y = \frac{1}{4} (1 - e^{- 4}) (e^{- 1} - e^{- 9})

= \frac{1}{4} {e^{- 1} - e^{- 9} - e^{- 5} + e^{- 13}} .

Answered by kaivan.ahmadi last updated on 02/Apr/19

\int_{1}^{3} \int_{0}^{2} {x y e}^{- x^{2} - y^{2}} d x d y = \frac{- 1}{2} \int_{1}^{3} {y e}^{- x^{2} - y^{2}} ∣_{0}^{2} d y =

\frac{- 1}{2} \int_{1}^{3} ({y e}^{- 4 - y^{2}} - {y e}^{- y^{2}}) d y =

\frac{- 1}{2} {(\frac{- 1}{2} e^{- 4 - y^{2}} + \frac{1}{2} e^{- y^{2}})}_{1}^{3} =

\frac{- 1}{4} [(e^{- 4 - 9} + e^{- 9}) - (e^{- 4 - 1} + e^{- 1})] =

\frac{- 1}{4} (e^{- 13} + e^{- 9} - e^{- 5} - e^{- 1})