Question

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

Short answer

The integral is approximately 3.2846.

Step-by-step solution

Determine the Interval Length

The given integral is evaluated from 2 to 4. Calculate the interval length, \( b - a \), where \( a = 2 \) and \( b = 4 \). This gives us an interval length of \( 4 - 2 = 2 \).

Calculate the Width of Each Subdivision

Since we are using four subdivisions, divide the interval length by 4. The width of each subdivision is \( \Delta x = \frac{2}{4} = 0.5 \).

Determine Midpoints of Each Subdivision

Determine the midpoints of each subdivision. For four intervals: 1. \( x_1 = 2 + 0.5 \times 0.5 = 2.25 \)2. \( x_2 = 2 + 1.5 \times 0.5 = 2.75 \)3. \( x_3 = 2 + 2.5 \times 0.5 = 3.25 \)4. \( x_4 = 2 + 3.5 \times 0.5 = 3.75 \)

Evaluate Function at Each Midpoint

Calculate the function value \( \frac{1}{\ln x} \) at each midpoint:1. \( f(2.25) = \frac{1}{\ln(2.25)} \)2. \( f(2.75) = \frac{1}{\ln(2.75)} \)3. \( f(3.25) = \frac{1}{\ln(3.25)} \)4. \( f(3.75) = \frac{1}{\ln(3.75)} \)

Apply the Midpoint Rule Formula

The midpoint rule formula is: \[ \int_{a}^{b} f(x) \, dx \approx \Delta x \sum_{i=1}^{n} f(m_i) \]Substitute \( \Delta x = 0.5 \) and the function values at the midpoints:\[ 0.5 \left( f(2.25) + f(2.75) + f(3.25) + f(3.75) \right) \]

Compute the Numerical Approximation

Calculate each term:1. \( f(2.25) = \frac{1}{\ln(2.25)} \approx 2.0825 \)2. \( f(2.75) = \frac{1}{\ln(2.75)} \approx 1.6738 \)3. \( f(3.25) = \frac{1}{\ln(3.25)} \approx 1.4705 \)4. \( f(3.75) = \frac{1}{\ln(3.75)} \approx 1.3424 \)Sum these values: \( 2.0825 + 1.6738 + 1.4705 + 1.3424 = 6.5692 \)Then multiply by \( \Delta x = 0.5 \):\( 0.5 \times 6.5692 = 3.2846 \)

Conclusion

Thus, the approximate value of \( \int_{2}^{4} \frac{1}{\ln x} \, dx \) using the midpoint rule with four subdivisions to four decimal places is \( 3.2846 \).

Key concepts

Numerical Integration

Numerical integration is a method used to approximate the value of integrals, especially when finding an exact solution analytically is challenging or impossible. This approach is essential for solving real-world problems involving complex functions.
There are various techniques within numerical integration, including the trapezoidal rule, Simpson's rule, and the midpoint rule. Each technique has its own strengths and is suitable for different types of functions and applications.
Numerical methods convert the integral of a function into a sum, allowing computers or calculators to handle the computations efficiently. By doing this, it is easier to obtain an estimated value for the equation that accurately reflects the function's behavior over a specific interval.
Understanding numerical integration helps in approaching problems where exact solutions are out of reach, providing a practical tool in science, engineering, and economics.

Midpoint Rule

The midpoint rule is a popular numerical integration technique used to approximate definite integrals. It specifically utilizes the midpoint of subintervals to estimate the area under a curve.
Here's how it works:

• Divide the entire interval of integration into smaller, equal subintervals.
• For each subinterval, identify the midpoint, which will be used as the representative point for that section.
• Evaluate the function at each of these midpoints.
• Multiply the function value by the width of the subinterval.

The formula for the midpoint rule can be expressed as:
\[ \int_{a}^{b} f(x) \, dx \approx \Delta x \sum_{i=1}^{n} f(m_i) \]
This formula allows you to estimate the integral by summing up the contributions from each subinterval.
The rule is particularly useful due to its balance between complexity and accuracy. While not as precise as some other numerical methods, it offers a good approximation with fewer calculations. It is often used when a simple, quick estimate is needed.

Definite Integral

A definite integral represents the area under a curve over a specific interval (\[ [a, b] \] ). It is a fundamental concept in calculus that measures accumulated quantities, such as total distance, area, or even volume.
The notation for a definite integral is \( \int_{a}^{b} f(x) \, dx \), where \( f(x) \) is the integrand, and \( a \) and \( b \) are the limits of integration that define the interval.
Definite integrals have many applications in real-world situations, including evaluating the total displacement of an object given its velocity over time, or calculating the work done by a force moving an object over a distance.
The result of a definite integral is always a real number, which makes it different from an indefinite integral, which represents a family of functions.

Approximation Methods

Approximation methods in calculus are crucial for estimating solutions to problems where exact answers are difficult or impossible to obtain directly. There are several well-known methods, including the midpoint rule, trapezoidal rule, and Simpson's rule.
These methods aim to provide values that are close to the true values of integrals or other operations. They are especially valuable in computational fields where dealing with complex functions analytically is challenging.

• The midpoint rule uses midpoints of subintervals to calculate approximate integrals.
• The trapezoidal rule considers function values at the endpoints to form trapezoids, which approximate the area under a curve.
• Simpson's rule applies parabolic segments instead of straight lines to provide more accurate estimates.

Each method involves a trade-off between computational complexity and accuracy. When selecting a method, one considers the function's behavior, required precision, and computational resources available.
Approximation methods form the backbone of numerical analysis and engineering calculations, making them indispensable in scientific research and everyday applications.