Evaluate-0-0-e-x-2-y-2-dydx-

Question Number 20612 by ajfour last updated on 29/Aug/17

E v a l u a t e \int_{0}^{\infty} \int_{0}^{\infty} e^{- (x^{2} + y^{2})} d y d x .

Answered by ajfour last updated on 29/Aug/17

l e t x^{2} + y^{2} = r^{2}

t h a t i s x = r \cos θ; y = r \sin θ

d x = - r \sin θ d θ; d y = r \cos θ d θ

\int_{0}^{\infty} \int_{0}^{\infty} e^{- (x^{2} + y^{2})} d y d x = I

I = \int_{0}^{\infty} [\int_{0}^{π / 2} e^{- r^{2}} r d θ] d r

= \frac{π}{2} \int_{0}^{\infty} e^{- r^{2}} r d r = \frac{π}{4} \int_{\infty}^{0} e^{- r^{2}} (- 2 r d r)

I = \frac{π}{4} (e^{- r^{2}}) ∣_{\infty}^{0} = \frac{π}{4} .

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