Prove that the Gaussian integral ∫₋∞^∞ e^(-x²) dx = √π using polar coordinates.
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Step-by-Step Solution
Step 1: Define the integral and square it
Let the given integral be I. To use polar coordinates, we need a two-dimensional integral. We can achieve this by squaring the integral. We write the second integral with respect to a different variable, y, which does not change its value.
Step 2: Combine into a double integral
Since the integrals are independent, we can combine the product of two single integrals into a single double integral over the entire xy -plane. The exponent becomes −(x2+y2).
Step 3: Convert to polar coordinates
Now, we convert the double integral from Cartesian coordinates to polar coordinates. We use the transformations x=rcosθ, y=rsinθ, which means x2+y2=r2. The differential area element dx dy becomes rdrdθ. The limits of integration for r are from 0 to ∞ (covering the entire plane), and for θ are from 0 to 2π.
Step 4: Evaluate the inner integral
We evaluate the inner integral with respect to r. We use a substitution u=r2, which implies du=2rdr. The limits of integration remain the same for u as for r. This simplifies the integral to a basic exponential form.
Step 5: Evaluate the outer integral and find I
Now we substitute the result of the inner integral back into the double integral and evaluate the outer integral with respect to θ. This gives us I2=π. Taking the square root, we find I=π. Since e−x2 is always positive, the integral I must be positive.