Question

Use the midpoint rule with \(n=2\) to estimate \(\int_{1}^{2} \frac{1}{x} d x\).

Short answer

The estimated integral is approximately 0.6857.

Step-by-step solution

Understand the Midpoint Rule

The midpoint rule is a numerical method used to approximate the definite integral \( \int_{a}^{b} f(x) \, dx \). When using \( n \) subintervals, the interval \( [a, b] \) is divided into \( n \) equal parts. The midpoint of each subinterval is used to estimate the area. For this exercise, we are using the midpoint rule with \( n = 2 \).

Determine the Width of Each Subinterval

To use the midpoint rule, first calculate the width of each subinterval, \( \Delta x \), by the formula: \( \Delta x = \frac{b-a}{n} \). Here, \( a = 1 \), \( b = 2 \), and \( n = 2 \). Thus, \( \Delta x = \frac{2-1}{2} = 0.5 \).

Identify Midpoints of Subintervals

Since \( \Delta x = 0.5 \), the endpoints of the subintervals are \( x_0 = 1.0 \), \( x_1 = 1.5 \), and \( x_2 = 2.0 \). The midpoints will be \( x_1^{*} = \frac{x_0 + x_1}{2} = \frac{1.0 + 1.5}{2} = 1.25 \) and \( x_2^{*} = \frac{x_1 + x_2}{2} = \frac{1.5 + 2.0}{2} = 1.75 \).

Evaluate the Function at Midpoints

The function given is \( f(x) = \frac{1}{x} \). Evaluate the function at each midpoint found in Step 3: \( f(1.25) = \frac{1}{1.25} \) and \( f(1.75) = \frac{1}{1.75} \). Simplifying, we have \( f(1.25) = 0.8 \) and \( f(1.75) \approx 0.5714 \).

Calculate the Approximate Integral

Apply the midpoint rule formula: \( M_n = \Delta x \sum_{i=1}^{n} f(x_i^{*}) \). Substituting the values from above, we have: \[ M_2 = 0.5 (f(1.25) + f(1.75)) = 0.5 (0.8 + 0.5714) = 0.5 \times 1.3714 = 0.6857 \]

Key concepts

Numerical Integration

Numerical integration is a mathematical technique used to approximate the value of an integral when direct analytical solutions are difficult or impossible. This method becomes particularly useful when dealing with definite integrals, or when the function you're integrating is complex or lacks a simple antiderivative. It provides a numerical value that is close to the exact integral.

There are several types of numerical integration methods, each fitting different types of problems and accuracy requirements. Some popular methods include:

• Midpoint Rule
• Trapezoidal Rule
• Simpson's Rule

These methods work by breaking down the area under a curve into simpler shapes (like rectangles or trapezoids) whose areas can be calculated easily.

The primary advantage of numerical integration is its versatility and applicability to a wide range of functions, including those that you cannot integrate analytically. However, the trade-off is that it provides an estimation rather than an exact answer. The accuracy of this estimation can often be improved by increasing the number of subintervals or choosing an appropriate method for the specific problem.

Definite Integral

A definite integral, denoted as \( \int_{a}^{b} f(x) \, dx \), represents the area under the curve of a function \( f(x) \) from point \( a \) to point \( b \) on the x-axis. Unlike indefinite integrals, which yield a family of functions, definite integrals calculate a specific numerical result.

In geometric terms,

• The definite integral can be visualized as the accumulation of small areas under the curve, between the limits \( a \) and \( b \).
• These areas can be positive or negative depending on whether the function is above or below the x-axis.

For continuous functions, the definite integral provides an exact measurement. However, it's solved through antiderivatives, which aren't always straightforward to find. This is where numerical methods come into play.

Numerical integration methods estimate this area using calculations on selected points within the curve. While numerical methods introduce a degree of approximation, they are a powerful tool, especially when an analytic solution is not possible. Understanding the properties and implications of definite integrals is crucial in fields ranging from physics to economics.

Approximation Methods

Approximation methods are techniques used in numerical integration to estimate integrals when exact solutions are not feasible. These methods simplify complex problems into more manageable calculations. The Midpoint Rule, one of these methods, estimates integrals by taking the value of the function at the midpoint of each subinterval and multiplying it by the width of the subinterval.

Some key aspects of approximation methods include:

• **Subintervals:** Dividing the region over which the integral is calculated into small sections.
• **Function Evaluation:** Estimating the value of the function often at a single point, or at a few strategic points within each subinterval.
• **Summation of Areas:** Calculating the area contribution of each subinterval and summing them to approximate the whole area under the curve.

The accuracy of approximation depends largely on:

• The number of subintervals \( n \): More subintervals generally lead to a more accurate approximation.
• The behavior of the function: Sudden changes in the function's behavior can lead to inaccuracies unless handled with a more sophisticated method.

Approximation methods are invaluable in scientific and engineering applications, where exact solutions are impractical or impossible to compute.