find-the-value-of-I-a-D-a-e-x-2-y-2-2-dxdy-with-D-a-x-y-R-2-x-2-y-2-a-2-

Question Number 27182 by abdo imad last updated on 02/Jan/18

f i n d t h e v a l u e o f I_{a} = \int \int_{D_{a}} e^{- \frac{x^{2} + y^{2}}{2}} d x d y w i t h

D_{a} = {(x, y) \in R^{2} / x^{2} + y^{2} ⩽ a^{2}}

Commented by abdo imad last updated on 05/Jan/18

w e u s e t h e p o l a r c o o r d i n a t e s a n d t h e d i f f e o m o r p h i s m e ψ

(r, θ) - > ψ (r, θ) = (x, y) = (r c o s θ, r s i n θ)

I_{a} = \int \int_{o ⩽ r ⩽ a} e^{- \frac{r^{2}}{2}} r d r d θ

= \int_{0}^{2 π} (\int_{0}^{a} r e^{- \frac{r^{2}}{2}} d r) d θ = 2 π \int_{0}^{a} r e^{- \frac{r^{2}}{2}} d r

= 2 π {[- e^{- \frac{r^{2}}{2}}]}_{0}^{a} = 2 π (1 - e^{- \frac{a^{2}}{2}})