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Ch. 8 - Integration Techniques
Briggs - Calculus: Early Transcendentals 3rd Edition
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Tutti i libri di testoBriggs 3rd EditionCh. 8 - Integration TechniquesProblema 8.8.44a
Capitolo 8, Problema 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).
fig
a. Estimate the surface area of the paneling using the Trapezoid Rule.

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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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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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Definition of the Definite Integral

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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Linearization
Pratica correlata
Domanda del libro di testo

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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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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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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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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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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81. Possible and impossible integrals

Let Iₙ = ∫ xⁿ e⁻ˣ² dx, where n is a nonnegative integer.

a. I₀ = ∫ e⁻ˣ² dx cannot be expressed in terms of elementary functions. Evaluate I₁.

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