Question

Use the function values in the following table and the Trapezoidal Rule with \(n=6\) to approximate \(\int_{2}^{8} f(x) d x\) $$\begin{array}{c|cccccc}{x} & {2} & {3} & {4} & {5} & {6} & {7} & {8} \\\ \hline f(x) & {16} & {19} & {17} & {14} & {13} & {16} & {20}\end{array}$$

Short answer

The approximation of the integral \(\int_{2}^{8} f(x) d x\) by the Trapezoidal Rule is 60.

Step-by-step solution

Understand the Trapezoidal Rule

The Trapezoidal Rule is synthesized by the following formula: \[ T(f) = \frac{h}{2} [f(x_0) +2f(x_1)+ 2f(x_2)+...+ 2f(x_{n-1}) + f(x_n)],\] where \(h\) is the width of each interval and is equal to \(h = (b-a)/n, n\) is the number of intervals, \(x_i = a + i*h\), and \(a, b\) are the lower and upper limits of the integration, respectively.

Compute the width of each interval

Calculate the width of each interval using the formula \(h = (b-a)/n\). Here, \(a = 2\), \(b = 8\) and \(n = 6\), so \(h = (8-2)/6 = 1\).

Calculate the result of integral

Apply the values to the formula mentioned in step 1, where \(h = 1\) and \(f(x)\) values are taken as given in the question \[ T(f) = \frac{1}{2} [(16 +2*19 +2*17 +2*14 +2*13 +2*16 + 20)] = 60.\] The output is the approximation of the integral \(\int_{2}^{8} f(x) d x\) by the Trapezoidal Rule and equals 60.

Key concepts

Numerical Integration

Numerical integration is a cornerstone in the field of calculus and it provides methods to approximate the value of a definite integral. When we have a function that is difficult to integrate analytically or when we are dealing with data points rather than a continuous function, numerical integration comes into play to provide an estimate for the integral.

One of the most common numerical integration techniques is the Trapezoidal Rule. This method involves partitioning the interval of integration into smaller segments and approximating the area under the curve by calculating the area of trapezoids that lie above each segment. The more segments (or trapezoids) used, the better the approximation tends to be. This technique is particularly useful for engineers and scientists who work with discrete data points and require quick and reliable estimates of an integral.

An understanding of numerical integration is essential for students and professionals who need to solve real-world problems that involve calculating areas, volumes, and other quantities represented by integrals.

Definite Integrals

A definite integral is a fundamental concept in calculus that represents the exact area under a curve, bounded by two specified points on the x-axis. It is notated as \( \int_{a}^{b} f(x) dx \), where \( f(x) \) is the function being integrated, and \( a \) and \( b \) are the lower and upper limits of integration, respectively.

The Trapezoidal Rule is used to approximate this defined area when finding the exact value analytically may be too challenging or impossible. By using this rule, as demonstrated in the provided exercise, students can learn to estimate the value of definite integrals by utilizing function values at specific points. This not only bolsters their understanding of integration but also instills an appreciation for the approximative nature of some mathematical procedures in real-world applications.

While understanding the concept of definite integrals is essential, it is equally important to recognize the applications of numerical methods like the Trapezoidal Rule which serve as practical tools for approximating these integrals.

Calculus

Calculus is a branch of mathematics focused on change and motion. It is divided into two main parts: differentiation and integration. Differentiation concerns rates of change and slopes of curves, while integration, which is the focus of the discussed Trapezoidal Rule, deals with the accumulation of quantities, such as areas under curves.

The Trapezoidal Rule bridges the concept of definite integrals with numerical methods, highlighting the practicality of calculus. This rule is a manifestation of how calculus can handle complex problems that are otherwise difficult to solve analytically. By integrating these approaches, students gain a more profound competence in calculus that empowers them to tackle a wide range of problems in science, engineering, economics, and beyond.

Calculus, with its rich concepts like numerical integration and definite integrals, remains essential for students in STEM fields. The understanding of these concepts is not just academically beneficial but also vital for solving practical issues in various industries and research domains.