Part 4: An Important Result in Statistics

Finally, the value of the integral

I = ó õ ¥ 0  e-x2dx

is very important in statistical applications. To evaluate it, we first notice that

I 2 = é ë ó õ ¥ 0  e-x2dx ù û é ë ó õ ¥ 0  e-y2dy ù û = ó õ ¥ 0  ó õ ¥ 0  e-x2e-y2dydx

That is, I ² is a type I iterated integral which can be converted to polar coordinates.       

EXAMPLE 6    Evaluate the integral

I 2 = ó õ ¥ 0  ó õ ¥ 0    e-x2e-y2dydx

Solution: To do so, let us notice that

I 2 = ó õ ó õ Quad  I      e-( x2+y2) dA

However, in polar coordinates, the first quadrant is given by r = 0 to r = ¥ for q = 0 to q = p/2. Thus,

I2 = ó õ p/2 0  ó õ ¥ 0  e-r2rdrdq

As a result, we can write

I2 = ó õ p/2 0  é ë lim R® ¥  ó õ R 0  e-r2rdr ù û dq

Thus, if we let u = r², du = 2rdr, u( 0) = 0, u(R) = R², then

I2 =  1 2 ó õ p/2 0  é ë lim R® ¥  ó õ R2 0  e-u du ù û dq =  1 2 ó õ p/2 0  é ë lim R® ¥  (e0-e-R2) ù û dq =  1 2 ó õ p/2 0  dq =  p 4

Thus, I = Öp /2, which implies both

ó õ ¥ 0  e-x2dx = p 2         and       ó õ ¥ -¥  e-x2dx =   p