A definite Double Integral!
Find
− ∞ ∫ ∞ − ∞ ∫ ∞ e − ( x 2 + y 2 ) d x d y
The answer is 3.14159.
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2 solutions
Cool! You could have also used the concept of Jacobians, by using polar coordinates substitutions for x and y. (I actually expected that as one of the solutions)
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Your wish is granted, Chandramouli…..6 years later! See my cylindrical coordinates solution above.
You can actually see its the modification of the pdf of a bivariate normal with means 0, sds 1/sqrt(2) each; and they are independent.
Let x 2 + y 2 = r 2 in polar form. Since we are integrating over the entire x y − plane, we require r ∈ [ 0 , ∞ ) , θ ∈ [ 0 , 2 π ] . Thus, the double integration transforms into:
∫ 0 2 π ∫ 0 ∞ r e − r 2 d r d θ = π .
I = ∬ − ∞ ∞ e − ( x 2 + y 2 ) d x d y
I = ∬ − ∞ ∞ e − x 2 e − y 2 d x d y
I = ∫ − ∞ ∞ e − y 2 d y ∫ − ∞ ∞ e − x 2 d x
I = ∫ − ∞ ∞ e − y 2 ( π ) d y
I = π ∫ − ∞ ∞ e − y 2 d y
I = π π = π = 3 . 1 4 1 5 9