Question

Using Simpson's rule with four subdivisions, find \(\int_{0}^{\pi / 2} \cos (x) d x\).

Short answer

The approximate value of the integral using Simpson's rule is 1.

Step-by-step solution

Identify the Interval and Subdivisions

We want to approximate the integral \(\int_{0}^{\pi/2} \cos(x) \, dx\) using Simpson's rule. The lower limit \(a\) is 0, the upper limit \(b\) is \(\pi/2\), and we have 4 subdivisions, which implies \(n = 4\).

Calculate the Step Size

The step size \(h\) is calculated as follows: \(h = \frac{b-a}{n} = \frac{\pi/2 - 0}{4} = \frac{\pi}{8}\).

List the Function Values

Since we have 4 intervals, we will list the 5 function values \(f(x)\) at points \(x_0, x_1, x_2, x_3, x_4\). Calculate each \(x_i\):- \(x_0 = 0\)- \(x_1 = \frac{\pi}{8}\)- \(x_2 = \frac{2\pi}{8} = \frac{\pi}{4}\)- \(x_3 = \frac{3\pi}{8}\)- \(x_4 = \frac{4\pi}{8} = \frac{\pi}{2}\)Now, calculate \(f(x_i) = \cos(x_i)\):- \(f(x_0) = \cos(0) = 1\)- \(f(x_1) = \cos(\frac{\pi}{8})\)- \(f(x_2) = \cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}\)- \(f(x_3) = \cos(\frac{3\pi}{8})\)- \(f(x_4) = \cos(\frac{\pi}{2}) = 0\).

Apply Simpson's Rule

Simpson's rule formula is:\[\int_{a}^{b} f(x) \, dx \approx \frac{h}{3}\left[f(x_0) + 4\sum_{\text{odd } i} f(x_i) + 2\sum_{\text{even } i} f(x_i) + f(x_n)\right]\]Substituting the values, we have:\[\frac{\pi}{24}\left[1 + 4\left(\cos\left(\frac{\pi}{8}\right) + \cos\left(\frac{3\pi}{8}\right)\right) + 2\cdot\frac{\sqrt{2}}{2} + 0\right]\]

Calculate and Simplify

Simplify and calculate the values:1. Calculate \(\cos(\frac{\pi}{8})\), \(\cos(\frac{3\pi}{8})\) using a calculator.2. Perform necessary multiplications and additions: - \(4(\cos(\frac{\pi}{8}) + \cos(\frac{3\pi}{8}))\).3. Combine all the terms.4. Multiply by \(\frac{\pi}{24}\) to obtain the final value.The computed approximate value of the integral is \(\approx 1\).

Key concepts

Numerical Integration

Numerical Integration is a method used to estimate the value of a definite integral. Sometimes, exact integration isn't feasible, particularly for complex functions or when we lack an analytical solution. Numerical methods come in handy for approximations.

One of the popular methods is Simpson's Rule, which provides an excellent balance of accuracy and ease of use. This rule fits parabolic arcs over intervals in the function, giving a smoother, more accurate approximation than other methods.

By considering more subdivisions or smaller intervals, the numerical approximation becomes more accurate, as it accounts for more subtle changes in the function's value across the domain.

Definite Integral

A Definite Integral refers to the evaluation of the integral of a function over a specified interval \([a, b]\). It is represented by \(\int_{a}^{b} f(x) \, dx\) and gives the net area between the function and the x-axis over that interval.

This concept is pivotal in understanding how areas under curves can be calculated, which can be translated to real-world scenarios such as finding displacement from velocity or total income from a varying rate function.

In this context, we're dealing with the integral of a trigonometric function, specifically \(\cos(x)\), from 0 to \(\pi/2\). Using Simpson's Rule here helps us find an accurate estimate without doing cumbersome algebra.

Trigonometric Functions

Trigonometric Functions, including sine, cosine, and tangent, describe relationships in triangles and periodic phenomena. These functions are essential in math, science, and engineering, often representing waves and cycles like sound and light.

In this problem, \(\cos(x)\) is the trigonometric function we're integrating. It has a periodic nature, meaning it repeats its values at regular intervals. This characteristic plays a crucial role in solving integration problems and is vital in understanding the behavior of waves and oscillations in different fields.

By evaluating \(\cos(x)\) at specific intervals during this exercise, we effectively capture the function's behavior, ensuring our Simpson's Rule approximation is as precise as possible.

Subdivisions

In numerical integration, Subdivisions refer to dividing the interval \([a, b]\) into smaller parts. These smaller segments allow us to apply numerical methods such as Simpson's Rule more effectively by using them to estimate the integral's value.

In our example with 4 subdivisions, each carries an equal length given by the step size \((h)\), calculated as \(\frac{b-a}{n}\). This approach balances accuracy and computational efficiency, as more subdivisions can lead to better approximations by capturing more of the underlying function's behavior.

The greater the number of subdivisions, the more data points are considered in the rule's formula, enhancing the estimate's reliability. However, each additional data point also adds to computational complexity, so a trade-off often needs to be made between accuracy and efficiency.