I need to express the following improper integral as a double integral of x and y and then, using polar coordinates, evaluate it.
I=∫∞−∞e−x2dx
Plotting it, we find a Gaussian centered at x=0 which tends to infinity to both sides. We can easily express it as a double integral :
I=∫10∫∞−∞e−x2dxdy
Evaluating both using Wolfram Alpha gives √π, so it converges.
I know that x=rcos(θ) and that dxdy=rdrdθ, but substituing this in the above integral and evaluating θ from 0 to 2π and r from 0 to ∞ doesn't yield the correct answer. What's wrong here?
Thanks a lot !
Answer
You could try:
I2=(∫∞−∞e−x2dx)(∫∞−∞e−y2dy)=∫∞−∞∫∞−∞e−(x2+y2)dx dy
then use the polar coordinates to compute the double integral.
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