The document discusses calculating the Gaussian integral, which is the integral of the Gaussian function over the entire real number line. It shows converting the integral to double integral form and then to polar coordinates. This allows evaluating the integral in closed form, arriving at the result that the Gaussian integral equals the square root of pi.

1. Gaussian Integral
John Carlo Agrado
John Carlo Gayola
BSAM -3A

2. Gaussian Integral
 The Gaussian integral, also known as the Euler–Poisson integral, is the integral of the Gaussian
function over the entire real line. Named after the
German mathematician Carl Friedrich Gauss, the integral is,

3. Evaluate
 Convert the integral to double Integral
 And Use Polar Cordinates

4.  Define the value of the integral to be I Then,
I=
 Rewrite this as Double Integral
 Instead X we are using Y
I²
Evaluate

5.  Multiply itself so we get,
I²=
Evaluate

6.  Convert to Polar Cordinates
When regarded as an integral on the plane, it is clear that we can regard x2 + y2 as just r2, and this
suggests we should convert the integral from Cartesian (x, y) to polar (r, θ) coordinates:
x²+y² = r²
dA=rdrdθ
I²= = 2π
Evaluate

7.  The last integral immediately suggests the substitution u = r2, giving
I²=
We conclude that,
Evaluate