Gaussian Integral Formula Recall

The standard form of the Gaussian integral is:

$ \int_{-\infin}^{\infin} e^{-ax^2} dx = \sqrt{\frac{\pi}{a}} $

This formula requires that $a > 0$. The integral $I$ given is:

$ I = \int_{-\infin}^{\infin} e^{\frac{-x^2}{b}} dx $

Relating Given Integral to Standard Form

Comparing the given integral $I$ with the standard form, we can see that:

$ a = \frac{1}{b} $

For the integral to converge, we must have $a > 0$, which implies $\frac{1}{b} > 0$, meaning $b$ must be positive ($b > 0$).

Calculating the Integral Value

Substituting $a = \frac{1}{b}$ into the standard formula, the value of the given integral $I$ is:

$ I = \sqrt{\frac{\pi}{(1/b)}} = \sqrt{\pi b} $

Solving for the Constant b

We are given that the value of the integral $I$ is 4. So, we set the expression for $I$ equal to 4:

$ \sqrt{\pi b} = 4 $

To find $b$, we first square both sides of the equation:

$ (\sqrt{\pi b})^2 = 4^2 $

$ \pi b = 16 $

Now, isolate $b$ by dividing both sides by $\pi$:

$ b = \frac{16}{\pi} $

Numerical Value and Rounding

Using the approximate value of $\pi \approx 3.14159265...$:

$ b \approx \frac{16}{3.14159265} $

$ b \approx 5.092958... $

Rounding the value of $b$ to two decimal places gives:

$ b \approx 5.09 $