Thursday, March 14, 2019

improper integrals - Evaluating intinftyinftyex2dx using polar coordinates.




I need to express the following improper integral as a double integral of x and y and then, using polar coordinates, evaluate it.



I=ex2dx



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=10ex2dxdy



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=(ex2dx)(ey2dy)=e(x2+y2)dx dy
then use the polar coordinates to compute the double integral.


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