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Conference rusure::math

Title:Mathematics at DEC
Moderator:RUSURE::EDP
Created:Mon Feb 03 1986
Last Modified:Fri Jun 06 1997
Last Successful Update:Fri Jun 06 1997
Number of topics:2083
Total number of notes:14613

488.0. "Something from Calculus 2" by PLDVAX::ZARLENGA () Tue May 13 1986 03:27

    	This is one from a Calculus final, I never did find out the
    'correct' way to do this ...
                                                       -x2
    	Evaluate the integral, from 0 to infinity of  e   dx

    mike zarlenga

        p.s. that's "e to the minus x squared".
T.RTitleUserPersonal
Name		DateLines
488.1Change of variables...ENGINE::ROTHTue May 13 1986 15:3132
    The standard trick is to take the integral of a product of two
    integrals of that form; since the integrand is even you can change
    the limits to -inf to +inf and divide by 2 when you're through.

    I = int(0, inf) exp(-x^2) dx

    I = 1/2 * int(-inf, inf) exp(-x^2) dx

    I^2 = 1/4 * int(-inf, inf) int(-inf, inf) exp(-x^2)*exp(-y^2) dx dy

    I^2 = 1/4 * int(-inf, inf) int(-inf, inf) exp(-(x^2+y^2)) dx dy

    Now the hack is to change to polar coordinates,

    dx dy = r dr dt,
    x^2 + y^2 = r^2,

    I^2 = 1/4 * int(0, 2*pi) int(0, inf) exp(-r^2) dr r dt,

    and you can do int(0, inf) exp(-r^2) r dr in closed form and get

    I^2 = 1/4 * int(0, 2*pi) * 1/2,

    so I = sqrt(pi)/2.

    You'll come across this integral in probability in connection with the
    Gaussian distribution;  the analysis can be extended to N dimensions to
    get the Chi-squared and other so called 'gamma' distributions.

    It also gives a nice formula for the volume of an N dimensional sphere.

- Jim