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Ch. 8 - Integration Techniques
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 8 - Integration TechniquesProblem 8.8.47c
Chapter 8, Problem 8.8.47c

45–48. {Use of Tech} Trapezoid Rule and Simpson’s Rule Consider the following integrals and the given values of n.
47. ∫(1 to e) (1/x) dx; n = 50
c. Compute the absolute errors in the Trapezoid Rule and Simpson’s Rule with 2n subintervals.

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Step 1: Understand the problem. You are tasked with computing the absolute errors in the Trapezoid Rule and Simpson’s Rule for the integral ∫(1 to e) (1/x) dx, using 2n subintervals where n = 50. This means you will use 100 subintervals for both methods.
Step 2: Recall the formulas for the Trapezoid Rule and Simpson’s Rule. The Trapezoid Rule approximates the integral as a sum of trapezoidal areas, while Simpson’s Rule uses parabolic segments for approximation. Both methods depend on the number of subintervals and the function values at specific points.
Step 3: Compute the exact value of the integral ∫(1 to e) (1/x) dx. The antiderivative of 1/x is ln(x), so the exact value of the integral is ln(e) - ln(1). Since ln(e) = 1 and ln(1) = 0, the exact value is 1.
Step 4: Apply the Trapezoid Rule and Simpson’s Rule with 100 subintervals. For the Trapezoid Rule, divide the interval [1, e] into 100 equal subintervals, calculate the function values at the endpoints of each subinterval, and use the formula for the Trapezoid Rule. For Simpson’s Rule, divide the interval into 100 subintervals, calculate the function values at the endpoints and midpoints, and use the Simpson’s Rule formula.
Step 5: Compute the absolute errors. The absolute error for each method is the difference between the exact value of the integral (1) and the approximated value obtained using the Trapezoid Rule and Simpson’s Rule. Compare the errors to assess the accuracy of each method.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Trapezoid Rule

The Trapezoid Rule is a numerical method for approximating the definite integral of a function. It works by dividing the area under the curve into trapezoids rather than rectangles, providing a better approximation. The formula involves calculating the average of the function values at the endpoints of each subinterval and multiplying by the width of the subintervals. This method is particularly useful when an exact integral is difficult to compute.
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Simpson’s Rule

Simpson’s Rule is another numerical integration technique that provides a more accurate approximation than the Trapezoid Rule by using parabolic segments instead of straight lines. It requires an even number of subintervals and combines the function values at the endpoints and midpoints of the intervals. The formula is based on fitting a quadratic polynomial to the function over each pair of subintervals, which enhances the accuracy of the approximation, especially for smooth functions.
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Absolute Error

Absolute error measures the difference between the exact value of an integral and the approximate value obtained using numerical methods like the Trapezoid or Simpson’s Rule. It is calculated as the absolute value of this difference, providing a straightforward way to assess the accuracy of the approximation. Understanding absolute error is crucial for evaluating the effectiveness of numerical methods and ensuring that the results meet the desired precision.
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Related Practice
Textbook Question

45–48. {Use of Tech} Trapezoid Rule and Simpson’s Rule Consider the following integrals and the given values of n.

48. ∫(0 to π/4) (1/(1 + x²)) dx; n = 64

c. Compute the absolute errors in the Trapezoid Rule and Simpson’s Rule with 2n subintervals.

Textbook Question

82. A family of exponentials The curves y = x * e^(-a * x) are shown in the figure for a = 1, 2, and 3.

c. Find the area of the region bounded by y = x * e^(-a * x) and the x-axis on the interval [0, b]. Because this area depends on a and b, we call it A(a, b).

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Textbook Question

Gaussians An important function in statistics is the Gaussian (or normal distribution, or bell-shaped curve), f(x) = e^(-ax²).

c. Complete the square to evaluate ∫ from -∞ to ∞ of e^(-(ax² + bx + c)) dx, where a > 0, b, and c are real numbers.

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Textbook Question

43. A hot-air balloon is launched from an elevation of 5400 ft above sea level. As it rises, the vertical velocity is computed using a device (called a variometer) that measures the change in atmospheric pressure. The vertical velocities at selected times are shown in the table (with units of ft/min).

c. A polynomial that fits the data reasonably well is:

g(t) = 3.49t³ - 43.21t² + 142.43t - 1.75

Estimate the elevation of the balloon after five minutes using this polynomial.

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Textbook Question

Computing areas On the interval [0,2], the graphs of f(x)=x²/3 and g(x)=x²(9−x²)^(-1/2) have similar shapes.

c. Which region has greater area?

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Textbook Question

77. Tabular integration Consider the integral ∫ f(x)g(x) dx, where f can be differentiated repeatedly and g can be integrated repeatedly

Let Gₖ represent the result of calculating k indefinite integrals of g (omitting constants of integration).

d. The tabular integration table from part (c) is easily extended to allow for as many steps as necessary in the process of integration by parts.

Evaluate ∫ x² e^(x/2) dx by constructing an appropriate table, and explain why the process terminates after four rows of the table have been filled in.

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