Question

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

Short answer

The estimate is approximately 0.70835.

Step-by-step solution

Identify the Interval

The integral \( \int_{1}^{2} \frac{1}{x} \, dx \) is defined over the interval \([1, 2]\). In this step, we set up the endpoints as \(a = 1\) and \(b = 2\).

Determine the Number of Subintervals

Given \(n = 2\), divide the interval \([1, 2]\) into 2 equal subintervals. Calculate the width of each subinterval using \(h = \frac{b-a}{n} = \frac{2-1}{2} = 0.5\).

Calculate Subinterval Points

The subinterval points are calculated as follows: \(x_0 = 1\), \(x_1 = 1 + 0.5 = 1.5\), and \(x_2 = 2\). These points divide the interval into two subintervals: \([1, 1.5]\) and \([1.5, 2]\).

Evaluate the Function at the Points

The function \(f(x) = \frac{1}{x}\) is evaluated at the subinterval points: \(f(x_0) = \frac{1}{1} = 1\), \(f(x_1) = \frac{1}{1.5} \approx 0.6667\), and \(f(x_2) = \frac{1}{2} = 0.5\).

Apply the Trapezoidal Rule

The trapezoidal rule is given by \(T_n = \frac{h}{2} [f(x_0) + 2f(x_1) + f(x_2)]\). Substitute known values: \(T_2 = \frac{0.5}{2} [1 + 2(0.6667) + 0.5]\).

Simplify the Expression

Compute \(T_2 = 0.25 [1 + 1.3334 + 0.5] = 0.25 \times 2.8334\). Further simplify: \(T_2 = 0.70835\).

Complete the Estimate

Therefore, the estimate of the integral \( \int_{1}^{2} \frac{1}{x} \, dx \) using the trapezoidal rule with \(n=2\) is approximately \(0.70835\).

Key concepts

Numerical Integration

Numerical integration is a powerful tool used to approximate the value of definite integrals, particularly when finding an exact solution analytically is complex or impossible. In simplest terms, numerical integration breaks down an integral into smaller, more manageable parts to estimate the whole. There are various methods for numerical integration. Each method employs a different strategy for estimating the area under a curve.

Some common techniques include:

• Trapezoidal Rule: This method approximates the integral by dividing the region into trapezoids instead of rectangles, offering more accuracy as it captures the slope of the curve better.
• Simpson's Rule: This technique provides an even more accurate estimation by using parabolic (curved) segments instead of straight-line segments.

These methods rely on evaluating the function at specific points across the interval. They then combine these values in a formula that sums up the approximate area under the curve. Understanding numerical integration can be extremely useful in fields such as physics, engineering, and finance. It offers a way to tackle real-world problems where precise analytical solutions may not be available.

Definite Integral

A definite integral represents the accumulation of quantities, like area, volume, or other physical properties, over a certain interval. The mathematical symbol for the definite integral is \( \int_a^b f(x)\,dx \), where \( a \) and \( b \) are the lower and upper bounds of the integration, respectively.

The main features of a definite integral include:

• Interval of Integration: The range \([a, b]\) represents the interval over which the function is integrated. In our exercise, this range is \([1, 2]\).
• Function to Integrate: The function \( f(x) \) under the integral sign is what you'll be summing up across the interval. In the example problem, this function is \( \frac{1}{x} \).

Definite integrals are essential in calculus because they provide a way to calculate the total accumulation or net change over an interval. This makes them highly applicable in calculating things like total growth or decline, balancing accounts over time, or determining the total distance covered given a velocity function.

Subintervals

Subintervals in numerical integration delineate the segments into which the entire interval is divided. This is a critical step, as the number and width of these subintervals can significantly influence the accuracy of the numerical approximation.

Key aspects of subintervals include:

• Number of Subintervals (\( n \)): This refers to how many equal parts the entire interval \([a, b]\) is divided into. More subintervals usually improve the estimate's accuracy, as you are taking into account more sections of the curve.
• Width of Subintervals (\( h \)): Calculated by \( h = \frac{b-a}{n} \), this width represents the length of each segment. In the given example, with \( n = 2 \), we find \( h = 0.5 \).

By computing the function values at the subinterval endpoints or midpoints, you can apply an appropriate rule (like the trapezoidal rule) to estimate the integral. Subdividing the domain helps in approximating the area under curves that cannot be integrated easily using only algebraic methods, thus serving as a bridge between pure mathematics and applied problems.