calculate-D-e-x-2-y-2-x-2-y-2-z-2-dxdydz-with-D-x-y-z-R-3-0-x-1-1-y-2-and-2-z-3-

Question Number 62213 by maxmathsup by imad last updated on 17/Jun/19

c a l c u l a t e \int \int \int_{D} e^{- x^{2} - y^{2}} \sqrt{x^{2} + y^{2} + z^{2}} d x d y d z w i t h

D = {(x, y, z) \in R^{3} / 0 ⩽ x ⩽ 1, 1 ⩽ y ⩽ 2 a n d 2 ⩽ z ⩽ 3}

Commented by prof Abdo imad last updated on 20/Jun/19

l e t u s e t h e d i f f e o m o r p h i s m x = r c o s θ, y = r s i n θ

0 ⩽ x ⩽ 1 a n d 1 ⩽ y ⩽ 2 \Rightarrow 1 ⩽ x^{2} + y^{2} ⩽ 5 \Rightarrow 1 ⩽ r ⩽ \sqrt{5}

\int \int \int_{D} e^{- x^{2} - y^{2}} \sqrt{x^{2} + y^{2} + z^{2}} d x d y d z

= \int_{2}^{3} (\int_{1}^{\sqrt{5}} (e^{- r^{2}} \sqrt{r^{2} + z^{2}}) r d r \int_{0}^{\frac{π}{2}} d θ) d z

= \frac{π}{2} \int_{2}^{3} (\int_{1}^{\sqrt{5}} ({r e}^{- r^{2}} \sqrt{r^{2} + z^{2}} d r) d z b u t

b y p a r t s

\int_{1}^{\sqrt{5}} ({r e}^{- r^{2}}) \sqrt{r^{2} + z^{2}} d r

= {[- \frac{1}{2} e^{- r^{2}} \sqrt{r^{2} + z^{2}}]}_{1}^{\sqrt{5}} + \frac{1}{2} \int_{1}^{\sqrt{5}} e^{- r^{2}} \frac{2 r}{\sqrt{r^{2} + z^{2}}} d r

= \frac{1}{2} (e^{- 1} \sqrt{1 + z^{2}} - e^{- 5} \sqrt{5 + z^{2}}) + \int_{1}^{\sqrt{5}} \frac{{r e}^{- r^{2}}}{\sqrt{r^{2} + z^{2}}} d r

r = z u \Rightarrow

\int_{1}^{\sqrt{5}} \frac{r e^{- r^{2}}}{\sqrt{r^{2} + z^{2}}} d r = \int_{\frac{1}{z}}^{\frac{\sqrt{5}}{z}} \frac{z u e^{- z^{2} u^{2}}}{z \sqrt{1 + u^{2}}} z d u

= z \int_{\frac{1}{z}}^{\frac{\sqrt{5}}{z}} \frac{u e^{- z^{2} u^{2}}}{\sqrt{1 + u^{2}}} d u \dots . . b e c o n t i n u e d \dots