Evaluate $\int \int_{R} e^{- (x^{2} + y^{2})} d x d y$ , where $R$ is the region bounded by the circle $x^{2} + y^{2} = a^{2}$ .

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Solution

The integral can be evaluated by transforming it into polar coordinates.
First we transform the coordinates and the domain of integration
$x^{2} + y^{2} = a^{2} \Rightarrow x = r cos θ, y = r sin θ \Rightarrow (r cos θ)^{2} + (r sin θ)^{2} = a^{2}$
$\Rightarrow r = a$
$\Rightarrow I = \int \int_{R} e^{- (x^{2} + y^{2})} d x d y = \int_{0}^{2 π} \int_{0}^{a} e^{- r^{2}} r d r d θ = 2 π \int_{0}^{a} r e^{- r^{2}} d r$
Now we substitute $z = r^{2} \Rightarrow d z = 2 r d r$
$I = π \int_{0}^{a^{2}} e^{- z} d z = π (1 - e^{- a^{2}})$

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