Ch. 8 - Integration Techniques

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo

Briggs 3rd Edition

Ch. 8 - Integration Techniques

Problema 8.8.3

Capitolo 8, Problema 8.8.3

3. Explain geometrically how the Trapezoid Rule is used to approximate a definite integral.

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The Trapezoid Rule is a numerical method used to approximate the value of a definite integral by dividing the interval of integration into smaller subintervals and approximating the area under the curve using trapezoids.

Start by dividing the interval [a, b] into n equal subintervals, each of width Δx = (b - a) / n. The points dividing the interval are x₀, x₁, x₂, ..., xₙ, where x₀ = a and xₙ = b.

For each subinterval [xᵢ, xᵢ₊₁], approximate the area under the curve by forming a trapezoid. The heights of the trapezoid are determined by the function values at the endpoints of the subinterval, f(xᵢ) and f(xᵢ₊₁).

The area of each trapezoid is calculated using the formula: Area = (1/2) * (base) * (sum of heights), which translates to: Area = (1/2) * Δx * [f(xᵢ) + f(xᵢ₊₁)].

Finally, sum up the areas of all trapezoids to approximate the integral: ∫[a, b] f(x) dx ≈ Δx * [(1/2)f(x₀) + f(x₁) + f(x₂) + ... + f(xₙ₋₁) + (1/2)f(xₙ)]. This geometric approach provides an estimate of the total area under the curve.

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Definite Integral

A definite integral represents the signed area under a curve defined by a function over a specific interval [a, b]. It quantifies the accumulation of quantities, such as area, volume, or total change, and is calculated using the Fundamental Theorem of Calculus. Understanding this concept is crucial for grasping how numerical methods, like the Trapezoid Rule, approximate these areas.

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05:43

Definition of the Definite Integral

Trapezoid Rule

The Trapezoid Rule is a numerical method used to estimate the value of a definite integral by dividing the area under the curve into trapezoids rather than rectangles. Each trapezoid's area is calculated using the average of the function values at the endpoints of the subintervals, multiplied by the width of the interval. This method provides a more accurate approximation than using rectangles, especially for functions that are linear or nearly linear over small intervals.

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Power Rules

Geometric Interpretation

The geometric interpretation of the Trapezoid Rule involves visualizing the area under a curve as a series of trapezoids formed between the function and the x-axis. By approximating the curve with these trapezoids, we can estimate the total area more accurately. This visualization helps in understanding how the method works and why it can yield better approximations compared to simpler methods like the Rectangle Rule.

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Geometric Sequences - Recursive Formula

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96. Challenge

Show that with the change of variables u = √tan x, the integral

∫ √tan x dx

can be converted to an integral amenable to partial fractions. Evaluate

∫[0 to π/4] √tan x dx.

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76–83. Preliminary steps The following integrals require a preliminary step such as a change of variables before using the method of partial fractions. Evaluate these integrals.

79. ∫ [sec t / (1 + sin t)] dt

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6. Using the trigonometric substitution x = 8 sec θ, where x ≥ 8 and 0 < θ ≤ π/2, express tan θ in terms of x.

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7–84. Evaluate the following integrals.

16. ∫ [1 / (x⁴ – 1)] dx

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77–86. Comparison Test Determine whether the following integrals converge or diverge.

78. ∫(from 0 to ∞) dx / (eˣ + x + 1)