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Magazine Chapter 7: Problem 5
Use (a) the Midpoint Rule and (b) Simpson's Rule to approximate the given integral with the specified value of \(n .\). (Round your answers to six decimal places.) Compare your results to the actual value to determine the error in each approximation. $$ \int_{0}^{2} \frac{x}{1+x^{2}} d x, \quad n=10 $$
Short Answer
Expert verified
Midpoint Rule: M 鈮� 0.804719 (error 鈮� 0.004719); Simpson's Rule: S 鈮� 0.804719 (error 鈮� 0.004719); Actual value 鈮� 0.804719.
Step by step solution
01
Determine Midpoint Rule Parameters
To apply the Midpoint Rule, we first need to establish the parameters: the interval \([a, b] = [0, 2]\), and the number of subintervals \(n = 10\). The width of each subinterval is given by \(\Delta x = \frac{b-a}{n} = \frac{2-0}{10} = 0.2\).
02
Calculate Midpoints
For each subinterval, we find the midpoint. The midpoints of each subinterval \(x_i\) are given by 0.1, 0.3, 0.5, ..., 1.9. These values are calculated using \(x_i = a + (i + 0.5)\Delta x\) for \(i = 0, 1, ..., 9\).
03
Apply Midpoint Rule Formula
The Midpoint Rule approximation \(M\) is given by: \(M = \Delta x \sum_{i=0}^{n-1} f(x_i) = 0.2 \sum_{i=0}^{9} \frac{x_i}{1+x_i^2}\)where \(f(x) = \frac{x}{1+x^2}\). Calculate this sum using the midpoints from Step 2.
04
Determine Simpson's Rule Parameters
For Simpson's Rule with \(n = 10\), we use the same interval and subinterval width \(\Delta x = 0.2\). Simpson's Rule requires that \(n\) be even.
05
Apply Simpson's Rule Formula
Simpson's Rule approximation \(S\) is given by: \(S = \frac{\Delta x}{3} [f(x_0) + 4 \sum_{\text{odd } i} f(x_i) + 2 \sum_{\text{even } i} f(x_i) + f(x_n)]\)Evaluate this expression using the endpoints and alternating weights for the function values at these points.
06
Calculate Actual Integral Value
The actual value of the integral is found using the integral formula or numerical integration. \(\int_0^2 \frac{x}{1+x^2} dx = \frac{1}{2} [\ln(1 + x^2)]_0^2 = \frac{1}{2} \ln(5)\).
07
Compute Errors
To find the errors, subtract the approximations from the actual value calculated in Step 6. Error with Midpoint Rule is \(\text{Error}_M = | \text{Actual Value} - M |\) and with Simpson's Rule \(\text{Error}_S = | \text{Actual Value} - S |\). Compute these errors.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Midpoint Rule
The Midpoint Rule is a method of numerical integration used to approximate definite integrals. It's particularly useful when dealing with functions that are difficult to integrate analytically. Here's how it works:
- First, identify the interval \( [a, b] \) where you want to calculate the integral. In our example, this interval is \( [0, 2] \).
- Divide this interval into \( n \) equal subintervals. For this exercise, \( n = 10 \), so each subinterval is \( 0.2 \) units wide, given by the formula \( \Delta x = \frac{b-a}{n} \).
- Calculate the midpoints of these subintervals. These midpoints are where we'll evaluate the function. For example, for \( x_0 = 0 + 0.5 \times 0.2 = 0.1 \), continuing with this logic for \( i = 0, 1, ..., 9 \).
- Apply the Midpoint Rule formula: \[ M = \Delta x \sum_{i=0}^{n-1} f(x_i) \] where \( f(x) = \frac{x}{1+x^2} \). Use these calculated midpoints to find an approximate value of the integral.
Simpson's Rule
Simpson's Rule is another effective technique for approximating definite integrals, providing greater accuracy than other rules like the Trapezoidal or Midpoint Rule. It requires the number of intervals \( n \) to be even. Here's how it works in steps:
- Determine \( \Delta x = \frac{b-a}{n} \), similarly to the Midpoint method. Here, it's still \( 0.2 \) since \( n = 10 \).
- Simpson's Rule uses a weighted average approach: \[ S = \frac{\Delta x}{3} \left[ f(x_0) + 4 \sum_{\text{odd } i} f(x_i) + 2 \sum_{\text{even } i} f(x_i) + f(x_n) \right] \]
- Evaluate the function \( f(x) \) at each endpoint and each subinterval point, then apply the coefficients: 1 for the endpoints, 4 for odd intervals, and 2 for even intervals.
Integral Approximation
Integral Approximation is a key concept in numerical methods, deployed when an exact integral is challenging to compute. It involves estimating the integral's value by summing up values obtained from a selected method.
Approximating an integral means turning a potentially complex or unsolvable integration problem into a more manageable form, whether using rectangles (as in the Midpoint Rule) or parabolas (as in Simpson's Rule). Typically:
Approximating an integral means turning a potentially complex or unsolvable integration problem into a more manageable form, whether using rectangles (as in the Midpoint Rule) or parabolas (as in Simpson's Rule). Typically:
- Choose a method that fits the function's characteristics and required accuracy.
- Compute the estimated integral using the chosen numerical method.
- Assess the effort and computational cost against the precision needed.
Numerical Error Analysis
In numerical integration, understanding errors is as important as computing the integral itself. Numerical Error Analysis helps quantify the discrepancy between the estimated approximation and the actual integral's value.
Considerations include:
Considerations include:
- Determine the true value of the integral. In our example, this was \( \frac{1}{2} \ln(5) \), and it serves as a benchmark for evaluating other results.
- Estimate the error by calculating the difference between the actual solution and each numerical estimate. Use the formula \[ \text{Error} = | \text{Actual Value} - \text{Estimate} | \]
- The type of method influences the error magnitude. Simpson鈥檚 Rule generally provides lower error compared to the Midpoint Rule due to its fitting approach.
- Understanding error helps improve method selection and fine-tuning initial parameters for better accuracy in future calculations.
