Textbook Question
66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in the following exercises.
66. Let f(x) = cos(x²).
b. Calculate f''(x).
Ch. 8 - Integration Techniques
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 8, Problem 8.8.44a
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.
Verified step by step guidance1
Identify the points given on the curve: (0,0), (1,2.5), (2,3.2), (3,4), (4,6), (6,7), (7.5,3), and (8,0). These points represent the height (y-values) at specific positions (x-values) along the base.
Recall that the Trapezoid Rule is used to approximate the area under a curve by dividing the region into trapezoids. The formula for the Trapezoid Rule is:
\[\text{Area} \approx \sum_{i=1}^{n} \frac{(x_i - x_{i-1})}{2} (y_i + y_{i-1})\]
where \(x_i\) and \(x_{i-1}\) are consecutive x-values, and \(y_i\) and \(y_{i-1}\) are the corresponding y-values.
Calculate the width of each trapezoid by subtracting consecutive x-values: for example, \(1 - 0 = 1\), \(2 - 1 = 1\), \(3 - 2 = 1\), \(4 - 3 = 1\), \(6 - 4 = 2\), \(7.5 - 6 = 1.5\), and \(8 - 7.5 = 0.5\).
For each trapezoid, compute the area using the formula:
\[\text{Area}_i = \frac{(x_i - x_{i-1})}{2} (y_i + y_{i-1})\]
where \(y_i\) and \(y_{i-1}\) are the heights at the endpoints of the interval.
Sum all the trapezoid areas calculated in the previous step to get the total estimated surface area of the paneling.
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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 used to approximate the definite integral of a function. It works by dividing the area under a curve into trapezoids rather than rectangles, then summing their areas. This method improves accuracy over simple rectangular approximations, especially when the function is relatively smooth.
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Power Rules
Definite Integral as Area Under a Curve
The definite integral of a function over an interval represents the net area between the curve and the x-axis. In this problem, estimating the surface area of the paneling involves calculating the area under the curve defined by the given points, which can be approximated using numerical integration techniques like the Trapezoid Rule.
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Using Discrete Data Points for Approximation
When a function is known only at discrete points, numerical methods like the Trapezoid Rule use these points to approximate integrals. The given coordinates represent the curve's shape, allowing the calculation of trapezoid areas between consecutive points to estimate the total area under the curve.
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Related Practice
Textbook Question
85. Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. More than one integration method can be used to evaluate ∫ (1 / (1 - x²)) dx.
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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?
Textbook Question
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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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
Textbook Question
66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in the following exercises.
67. Let f(x) = √(x³ + 1).
a. Find a Midpoint Rule approximation to ∫[1 to 6] √(x³ + 1) dx using n = 50 subintervals.