Question

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

The approximation of the integral can be calculated by substituting the values into the equation derived from the Trapezoidal rule.

Step-by-step solution

Identify the function to be integrated

The function to be integrated is \[f(t)=\frac{1}{\sqrt{2 \pi}} \exp \left\{-\frac{1}{2} t^{2}\right\}\]

Identify the limits of integration

The limits of integration are from \(0\) to \(1.96\).

Decide on a numerical integration method

A simple method for numerical integration is the Trapezoidal rule. This rule averages the function values at the start and end points of the interval, and multiplies by the interval width to give an estimate for the integral over that interval.

Apply the numerical integration method

The trapezoidal rule is given as follows: \[\int_{a}^{b} f(x) d x \approx \frac{b-a}{2}\left[f(a)+f(b)\right]\] Use the function and the limits of integration to calculate the approximation as such: \[\int_{0}^{1.96} \frac{1}{\sqrt{2 \pi}} \exp \left\{-\frac{1}{2} t^{2}\right\} dt \approx \frac{1.96-0}{2}\left[\frac{1}{\sqrt{2 \pi}} \exp \left\{-\frac{1}{2} (0)^{2}\right\} + \frac{1}{\sqrt{2 \pi}} \exp \left\{-\frac{1}{2} (1.96)^{2}\right\}\right]\]

Simplify the expression

Simplify the expression to get the approximation value.

Key concepts

Trapezoidal Rule

The Trapezoidal Rule is a method of numerical integration used to approximate definite integrals. It is particularly useful when dealing with functions that do not have an elementary antiderivative, making analytical integration challenging. This method works by dividing the area under a curve into trapezoids rather than rectangles (as in the midpoint or left-point rule).

The formula for the trapezoidal rule when applied to function f(x) over the interval [a, b] is:\[\begin{equation}\int_{a}^{b} f(x) dx \approx \frac{b-a}{2}\left( f(a) + f(b) \right)\end{equation}\]To improve the accuracy of the trapezoidal rule, one can increase the number of trapezoids by dividing the interval into smaller subintervals. This method is very effective for approximating the area under smooth curves and works best when the function is approximately linear over each subinterval.For more complex functions with curvature, the trapezoidal rule becomes less accurate over wide intervals. Therefore, if high accuracy is desired, it is crucial to use a fine partition of the interval or opt for more sophisticated numerical integration methods such as Simpson's rule.

Definite Integrals

Definite integrals are a cornerstone of calculus, representing the net area under a function's graph on a specified interval. This concept extends beyond mere geometry, linking to numerous applications across the sciences. For instance, it’s used in physics for calculating work done by a variable force, in economics for determining consumer and producer surplus, and in probability theory for finding the likelihood of events.

The analytical approach requires finding an antiderivative F(x) of a function f(x), then evaluating it at the endpoints of the given interval [a, b] using the Fundamental Theorem of Calculus:\[\begin{equation}\int_{a}^{b} f(x) dx = F(b) - F(a)\end{equation}\]However, in cases where finding an antiderivative is difficult or impossible, numerical methods such as the Trapezoidal Rule, Simpson's Rule, and Riemann Sums are employed to approximate the value of a definite integral.

Normal Distribution

The normal distribution, a bell-shaped curve, is pivotal in statistics, underpinning the central limit theorem and the concept of standard deviation. It describes the spread of a set of data where most values cluster around a central mean value with probabilities for values further from the mean tapering off symmetrically.

Mathematically, the probability density function (pdf) of the normal distribution is given by:\[\begin{equation}f(x) = \frac{1}{\sqrt{2\pi}\sigma} \exp\left(-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^{2}\right)\end{equation}\]where \( \mu \) is the mean and \( \sigma \) is the standard deviation of the distribution. The integral of the pdf over all possible values (from \(-\infty\) to \(\infty\)) is equal to 1, which signifies the total probability.In our exercise, we are dealing with a section of the normal distribution curve, which requires calculating the area under the curve from 0 to 1.96. This area corresponds to the cumulative probability function for values up to 1.96 standard deviations above the mean. Such probabilities are essential for hypothesis testing and confidence interval estimation in statistics. The numerical integration, specifically using the Trapezoidal Rule, allows us to approximate this area when the corresponding integral cannot be computed analytically due to the lack of a simple antiderivative for the normal distribution's pdf.