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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 8.8.69a
Chapter 8, Problem 8.8.69a

66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in the following exercises.
69. Let f(x) = sin(eˣ).
a. Find a Trapezoid Rule approximation to ∫[0 to 1] sin(eˣ) dx using n = 40 subintervals.

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Identify the integral to approximate: \(\int_0^1 \sin(e^x) \, dx\) and note that we will use the Trapezoid Rule with \(n = 40\) subintervals.
Calculate the width of each subinterval using the formula \(\Delta x = \frac{b - a}{n} = \frac{1 - 0}{40} = \frac{1}{40}\).
Determine the \(x\)-values at which to evaluate the function: \(x_i = a + i \Delta x = 0 + i \times \frac{1}{40}\) for \(i = 0, 1, 2, \ldots, 40\).
Evaluate the function \(f(x) = \sin(e^x)\) at each \(x_i\) to get \(f(x_i) = \sin(e^{x_i})\) for all \(i\) from 0 to 40.
Apply the Trapezoid Rule formula: \(T_n = \frac{\Delta x}{2} \left[f(x_0) + 2 \sum_{i=1}^{n-1} f(x_i) + f(x_n)\right]\), substituting the values of \(f(x_i)\) and \(\Delta x\) to set up the approximation.

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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 to approximate definite integrals by dividing the interval into subintervals and approximating the area under the curve as trapezoids. The sum of these trapezoid areas provides an estimate of the integral, improving accuracy as the number of subintervals increases.
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Power Rules

Partitioning the Interval and Subintervals

To apply the Trapezoid Rule, the integration interval [a, b] is divided into n equal subintervals, each of width Δx = (b - a)/n. The function values at the endpoints and subinterval points are used to calculate the trapezoid areas, making the choice of n crucial for the approximation's precision.
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Interval of Convergence

Error Estimation in the Trapezoid Rule

The error bound for the Trapezoid Rule depends on the second derivative of the function over the interval. Theorem 8.1 provides a formula to estimate the maximum error, helping to understand how close the approximation is to the true integral and guiding the choice of n for desired accuracy.
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Determining Error and Relative Error
Related Practice
Textbook Question

65. Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

a. To evaluate ∫ (4x⁶)/(x⁴ + 3x²) dx, the first step is to find the partial fraction decomposition of the integrand.

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Textbook Question

75. {Use of Tech} Oscillator displacements Suppose a mass on a spring that is slowed by friction has the position function:

s(t) = e⁻ᵗ sin t

a. Graph the position function. At what times does the oscillator pass through the position s = 0?

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Textbook Question

A piece of wood paneling must be cut in the shape shown in the figure.

The coordinates of several points on its curved surface are also shown (with units of inches).

a. Estimate the surface area of the paneling using the Trapezoid Rule.

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Textbook Question

66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in the following exercises.

70. Let f(x) = e^(-x²).

a. Find a Simpson's Rule approximation to the integral from 0 to 3 of e^(-x²) dx using n = 30 subintervals.

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Textbook Question

41-44. {Use of Tech} Nonuniform grids

Use the indicated methods to solve the following problems with nonuniform grids.

41. A curling iron is plugged into an outlet at time t = 0. Its temperature T in degrees Fahrenheit, assumed to be a continuous function that is strictly increasing and concave down on 0 ≤ t ≤ 120, is given at various times (in seconds) in the table.

a. Approximate (1/120)∫(0 to 120)T(t)dt in three ways using a left Riemann sum, using a right Riemann sum and using the Trapezoid Rule

Interpret the value of (1/120)∫(0 to 120)T(t)dt in the context of this problem.

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Textbook Question

42. Approximating integrals The function f is twice differentiable on (-∞, ∞). Values of f at various points on [0, 20] are given in the table.

a. Approximate ∫(0 to 120) f(x) dx in three way using a left Riemann sum, a right Riemann sum and the Trapezoid Rule

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