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Chapter 4: Problem 3

Similar to Exercise \(4.8 .2\) but now approximate \(\int_{0}^{1.96} \frac{1}{\sqrt{2 \pi}} \exp \left\\{-\frac{1}{2} t^{2}\right\\} d t\)

Short Answer

Expert verified
The approximate value of the integral is 0.9750021 * \( \frac{\sqrt{\pi}}{2} \) / \( \sqrt{2 \pi} \) = 0.9750021 / \( \sqrt{2} \)

Step by step solution

01

Identify the integral

First, identify the integral that needs to be evaluated, which is \( \int_{0}^{1.96} \frac{1}{\sqrt{2 \pi}} \exp \left\{-\frac{1}{2} t^{2}\right\} dt \)
02

Use the error function

Recognize that the given integral resembles the error function (erf), the standard Gaussian integral, with the formula \(erf(x) = \frac{2}{\sqrt{\pi}} \int_{0}^{x} e^{-t^{2}} dt\). If you divide both sides by \( \frac{2}{\sqrt{\pi}} \), you can rewrite the integral as: \( \int_{0}^{x} e^{-t^{2}} dt = erf(x) \cdot \frac{\sqrt{\pi}}{2} \). Now we see that the integral is equivalent to the error function multiplied by \( \frac{\sqrt{\pi}}{2} \)
03

Apply the limit

Putting the expression in the appropriate form, you have:\( \int_{0}^{1.96} e^{-t^{2}} dt = erf(1.96) \times \sqrt{\pi} \times \frac{1}{2} \). Now, calculate the erf(1.96), which is approximately 0.9750021, multiply it by \( \frac{\sqrt{\pi}}{2} \), and divide it by \( \sqrt{2 \pi} \) from the original integral, to get your final result.

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