Problem 4 - Applied Calculus

Question

For each definite integral: a. Approximate it "by hand," using trapezoidal approximation with \(n=4\) trapezoids. Round calculations to three decimal places. b. Evaluate the integral exactly using antiderivatives, rounding to three decimal places. c. Find the actual error (the difference between the actual value and the approximation). d. Find the relative error (the actual error divided by the actual value, expressed as a percent). \(\int_{1}^{3} \frac{1}{x} d x\)

Step-by-Step Solution

Answer

The trapezoidal approximation is 1.117, exact value is 1.099; actual error is 0.018, relative error is 1.638%.

Step 1: Compute Dimensions for Trapezoids

Divide the interval from 1 to 3 into 4 equal parts. The width of each trapezoid, denoted as \( \Delta x \), is calculated as follows:\[\Delta x = \frac{3 - 1}{4} = \frac{1}{2} = 0.5\]

Step 2: Calculate Function Values at Key Points

Calculate the function values \( f(x) = \frac{1}{x} \) at the points \( x_0 = 1 \), \( x_1 = 1.5 \), \( x_2 = 2 \), \( x_3 = 2.5 \), and \( x_4 = 3 \):- \( f(1) = 1 \)- \( f(1.5) = \frac{2}{3} = 0.667 \)- \( f(2) = \frac{1}{2} = 0.5 \)- \( f(2.5) = \frac{2}{5} = 0.4 \)- \( f(3) = \frac{1}{3} = 0.333 \)

Step 3: Apply Trapezoidal Rule

The trapezoidal approximation is calculated using the formula:\[T \approx \frac{\Delta x}{2} \times (f(x_0) + 2f(x_1) + 2f(x_2) + 2f(x_3) + f(x_4))\]Substituting in the values:\[T \approx \frac{0.5}{2} \times (1 + 2(0.667) + 2(0.5) + 2(0.4) + 0.333)\]\[T \approx 0.25 \times (1 + 1.334 + 1 + 0.8 + 0.333)\]\[T \approx 0.25 \times 4.467 = 1.117\]

Step 4: Evaluate the Integral Exactly

The integral \( \int_{1}^{3} \frac{1}{x} \, dx \) is equal to:\[\left[ \ln |x| \right]_{1}^{3} = \ln(3) - \ln(1) = \ln(3)\]Rounding \( \ln(3) = 1.099 \) (approximated to three decimal places).

Step 5: Calculate the Actual Error

The actual error is the difference between the exact integral value and the trapezoidal approximation:\[|1.099 - 1.117| = 0.018\]

Step 6: Calculate the Relative Error

The relative error as a percentage is calculated by dividing the actual error by the exact value:\[\text{Relative Error} = \frac{0.018}{1.099} \times 100\%\]\[\approx 1.638\%\]

Key Concepts

Trapezoidal Rule

The Trapezoidal Rule is a simple yet effective numerical integration method used to approximate the value of a definite integral. It does this by dividing the area under the curve into a series of trapezoids instead of rectangles, for better accuracy. For the given integral \( \int_{1}^{3} \frac{1}{x} \, dx \), the interval \([1, 3]\) is split into 4 equal parts. Each sub-interval forms a trapezoid, with the width denoted by \( \Delta x = 0.5 \). By calculating the function's values at the specified points \( x = 1, 1.5, 2, 2.5, 3 \), we acquire heights of the trapezoids. These are substituted into the trapezoidal formula:

• \( T \approx \frac{\Delta x}{2} \times (f(x_0) + 2f(x_1) + 2f(x_2) + 2f(x_3) + f(x_4)) \)

This produces a result of approximately 1.117 when rounded to three decimal places. The primary advantage of the trapezoidal approximation over the rectangular approach is that it accounts for variation across the interval, offering a more accurate reflection of the area under the curve.

Antiderivatives

Antiderivatives are a fundamental concept in calculus, helping us find the exact value of definite integrals. They are essentially the inverse process of differentiation. To find the antiderivative of a function is to determine another function whose derivative is the original function.For the integral \( \int_{1}^{3} \frac{1}{x} \, dx \), the antiderivative of \( \frac{1}{x} \) is \( \ln|x| \). We calculate the definite integral by evaluating \( \ln|x| \) from 1 to 3:

• \( \int_{1}^{3} \frac{1}{x} \, dx = [\ln|x|]_1^3 = \ln(3) - \ln(1) \)
• Since \( \ln(1) = 0 \), it simplifies to \( \ln(3) \).

This antiderivative approach gives an exact integral value of approximately 1.099 when rounded to three decimal places.

Relative Error Calculation

Relative error calculation is essential in determining how accurate an approximation is in numerical integration methods. It is the ratio of the actual error to the exact value of the integral, expressed as a percentage. For the integral \( \int_{1}^{3} \frac{1}{x} \, dx \), the actual error, or the difference between the trapezoidal approximation and the true integral value, is calculated as:

• \( |1.099 - 1.117| = 0.018 \)

The relative error percentage is found by dividing the actual error by the exact integral value, then multiplying by 100:

• \( \text{Relative Error} = \frac{0.018}{1.099} \times 100 \approx 1.638\% \)

This will indicate how close the approximation is to the actual value, allowing us to gauge the effectiveness of the trapezoidal rule in this context.

Numerical Integration Methods

Numerical integration methods like the Trapezoidal Rule are powerful tools in calculus, particularly useful for approximating definite integrals when an exact solution is difficult or impossible to find analytically. These methods are based on numerical approximations and are essential for solving real-world problems in engineering, physics, and other sciences, where analytical integration might not be feasible. Some common methods include:

• Trapezoidal Rule: Uses trapezoids over subintervals for approximation.
• Simpson's Rule: Employs parabolic segments to approximate the area under the curve, often providing more accuracy.
• Midpoint Rule: Utilizes the midpoint of each subinterval to approximate the value of the integral.

Each method has its strengths and level of complexity, making them suitable for different types of functions and accuracy requirements. Understanding these methods allows us to choose the most appropriate one for a given problem, balancing precision and computational efficiency.