Integrate e^(-u^2) du: Polar Coords Explained

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To integrate e^(-u^2) du, it can be expressed as a double integral of e^(-x^2) dx and e^(-y^2) dy. The discussion emphasizes the need to convert Cartesian coordinates to polar coordinates for easier integration. The transformation involves using r and θ, where x = r cos(θ) and y = r sin(θ), and adjusting the area element accordingly. The limits of integration in polar coordinates need clarification, particularly how they relate to the area being integrated. Ultimately, the goal is to evaluate the integral to find the result, which is expected to be √π.
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Homework Statement



I need to integrate e^(-u^2) du


I know I can write this as a double integral of exp(-x^2)dx . exp(-y^2)dy

I'm told I need to change this into polar coordinates but I'm unsure really what's going on
 
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It may help if you describe what you are trying to accomplish. I don't follow how \int {e^{-{u^2}} du = \int\int e^{-{x^2}} e^{-{y^2}} dx dy. I think you are trying to say that it is the product of two integrals, \int {e^{-{u^2}} du = \int e^{-{x^2}} dx \int e^{-{y^2}} dyYou can't directly integrate e^{x^2}. I believe you can use a Maclaurin series to approximate a value. Look at the Maclaurin series for e^x and substitute x with x^2.

Assuming that your x and y are Cartesian, you can substitute their polar alternative. Just like on the unit circle, the x-horizontal axis component of a point is normally associated as the product of the magnitude of the point from the zero origin and the cosine of the angle, namely r \cos{\Theta}. Use this in your integral and you might get something that can be integrated, but without more about the problem, this may not be correct.
 
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Go to Wolframalpha.com and use their computational engine to integrate E^(-x^2) dx

Their solution is .5*Sqrt(Pi) erf(x) + Constant
 
the answer I am looking to get is SQRT(Pi).

The x and y were in cartesian yes. I have a part solution which states

"Change to circular polar coordinates (x, y) -> (rho,phi ) and evaluate the angular
integral (remember if you change variables correctly, the area of a ring should
enter as 2*Pi*r*dr)."

If r = (x,y) = (rho.cos(phi), rho.sin(phi) ), I am unsure of the limits or really what's going on. i don't see why you need cylindrical polars? (apart from the fact you can't do it in cartesian)
 
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