Question

Approximate \(\int_{2}^{4} \frac{1}{\ln x} d x\) using the trapezoidal rule with eight subdivisions to four decimal places.

Short answer

The approximate value of the integral is 3.7829.

Step-by-step solution

Identify the Function and Interval

The given integral to approximate is \( \int_{2}^{4} \frac{1}{\ln x} \, dx \). We need to approximate this using the trapezoidal rule with 8 subdivisions on the interval \([2, 4]\).

Calculate the Width of Subdivisions

The interval \([2, 4]\) is divided into 8 equal parts. To find the width \( h \) of each subdivision, use the formula \( h = \frac{b-a}{n} \), where \( n = 8 \) is the number of subdivisions. Thus, \( h = \frac{4-2}{8} = \frac{2}{8} = 0.25 \).

Determine the Subdivision Points

The points on the interval are given by \( x_0 = 2, x_1 = 2.25, x_2 = 2.5, x_3 = 2.75, x_4 = 3, x_5 = 3.25, x_6 = 3.5, x_7 = 3.75, x_8 = 4 \). These are calculated by adding the width \( h = 0.25 \) to each subsequent point.

Evaluate the Function at Each Subdivision Point

Compute \( f(x) = \frac{1}{\ln x} \) at each subdivision point:- \( f(2) = \frac{1}{\ln 2} \)- \( f(2.25) = \frac{1}{\ln 2.25} \)- \( f(2.5) = \frac{1}{\ln 2.5} \)- \( f(2.75) = \frac{1}{\ln 2.75} \)- \( f(3) = \frac{1}{\ln 3} \)- \( f(3.25) = \frac{1}{\ln 3.25} \)- \( f(3.5) = \frac{1}{\ln 3.5} \)- \( f(3.75) = \frac{1}{\ln 3.75} \)- \( f(4) = \frac{1}{\ln 4} \).

Apply the Trapezoidal Rule Formula

The trapezoidal rule formula is given by: \[T_n = \frac{h}{2} \left(f(x_0) + 2 \sum_{i=1}^{n-1} f(x_i) + f(x_n)\right)\]Substitute \( h = 0.25 \) and the evaluated function values into the formula.

Calculate the Approximation

Using the function values from Step 4 and substituting them into the trapezoidal rule formula, compute the approximation:\[T_8 = \frac{0.25}{2} \left(f(2) + 2 \cdot (f(2.25) + f(2.5) + f(2.75) + f(3) + f(3.25) + f(3.5) + f(3.75)) + f(4)\right)\]Upon calculation, the approximation is approximately 3.7829.

Key concepts

Numerical Integration

Numerical integration is a technique used to estimate the value of a definite integral when an analytical solution is difficult or impossible to obtain. This approach is crucial in calculus, computer science, and engineering, where integrals need evaluation but do not have simple antiderivatives.
Techniques like the trapezoidal rule, Simpson's rule, and Monte Carlo integration are common methods for numerical integration. These techniques often involve approximating the area under a curve by dividing it into sections, known as subdivisions or intervals, and calculating the area of these sections to achieve an estimate.
The importance of numerical integration lies in its ability to provide solutions in real-world problems where exact solutions are either unknown or require excessive computation. This makes numerical integration a fundamental tool in fields that require approximation due to complex functional forms.

Subdivisions in Calculus

In the context of calculus and numerical integration, subdivisions refer to breaking down an integral's interval into smaller segments. This division allows for more manageable calculations, especially when using approximation methods like the trapezoidal rule.
For example, consider an interval from 2 to 4. Dividing this into 8 subdivisions means each segment has a width, which we denote as \( h \). This subdivision allows us to analyze the function's behavior more closely over smaller increments.
These points of subdivision are crucial because they are where the numerical method calculates the function's value. Then, these values are used to approximate the total area under the curve between the initial and final points of the interval. The more subdivisions you use, generally, the more accurate the approximation will become due to finer detail of the curve being captured.

Approximation Methods

Approximation methods are strategies to find close estimates to otherwise difficult mathematical solutions. In calculus, one such method is the trapezoidal rule, a form of numerical integration involving approximating a curve as a series of trapezoids.
The trapezoidal rule works by evaluating the function at established points called nodes. Between each pair of nodes, the curve is approximated by a straight line, forming trapezoids. The area of these trapezoids is then summed to give an approximate value of the integral.
To apply the trapezoidal rule, you need to know three things:

• The function to integrate
• The interval over which to integrate
• The number of subdivisions

The effectiveness of approximation methods, like the trapezoidal rule, largely depends on the function's behavior. For complex or highly curved functions, more subdivisions may enhance the accuracy but also increase computational effort. This balance between accuracy and efficiency is a key part of choosing an appropriate approximation method.