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Ch. 5 - Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 5 - IntegrationProblema 5.1.69a
Capitolo 5, Problema 5.1.69a

Approximating areas Estimate the area of the region bounded by the graph of ƒ(𝓍) = x² + 2 and the x-axis on [0, 2] in the following ways.
(a) Divide [0, 2] into n = 4 subintervals and approximate the area of the region using a left Riemann sum. Illustrate the solution geometrically.

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First, divide the interval [0, 2] into 4 equal subintervals. Since the length of the interval is 2, each subinterval will have width \(\Delta x = \frac{2 - 0}{4} = 0.5\).
Identify the left endpoints of each subinterval. These will be \(x_0 = 0\), \(x_1 = 0.5\), \(x_2 = 1.0\), and \(x_3 = 1.5\). Note that the left Riemann sum uses the function values at these points.
Evaluate the function \(f(x) = x^2 + 2\) at each left endpoint: calculate \(f(x_0)\), \(f(x_1)\), \(f(x_2)\), and \(f(x_3)\).
Multiply each function value by the width \(\Delta x\) to find the area of each rectangle: \(f(x_i) \times \Delta x\) for \(i = 0, 1, 2, 3\).
Sum all these rectangle areas to approximate the total area under the curve on [0, 2]: \(\text{Left Riemann Sum} = \sum_{i=0}^{3} f(x_i) \Delta x\). This sum represents the approximate area bounded by the graph and the x-axis.

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Riemann Sums

Riemann sums approximate the area under a curve by dividing the interval into subintervals and summing the areas of rectangles formed using function values at specific points. The left Riemann sum uses the left endpoint of each subinterval to determine the rectangle height, providing an estimate of the integral.
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Introduction to Riemann Sums

Partitioning the Interval

Partitioning involves dividing the interval [0, 2] into equal subintervals, here n = 4, to create smaller segments for approximation. Each subinterval has length Δx = (b - a)/n, which is essential for calculating the width of rectangles in the Riemann sum.
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Interval of Convergence

Function Evaluation at Endpoints

To compute the left Riemann sum, evaluate the function f(x) = x² + 2 at the left endpoints of each subinterval. These values determine the heights of the rectangles, which when multiplied by the subinterval width and summed, approximate the total area under the curve.
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Evaluating Composed Functions
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Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

(a) If ƒ is symmetric about the line 𝓍 = 2 , then ∫₀⁴ ƒ(𝓍) d𝓍 = 2 ∫₀² ƒ(𝓍) d𝓍.

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Mass from density A thin 10-cm rod is made of an alloy whose density varies along its length according to the function shown in the figure. Assume density is measured in units of g/cm. In Chapter 6, we show that the mass of the rod is the area under the density curve.

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Free fall On October 14, 2012, Felix Baumgartner stepped off a balloon capsule at an altitude of almost 39 km above Earth’s surface and began his free fall. His velocity in m/s during the fall is given in the figure. It is claimed that Felix reached the speed of sound 34 seconds into his fall and that he continued to fall at supersonic speed for 30 seconds. (Source: http://www.redbullstratos.com)

(a) Divide the interval [34, 64] into n = 5 subintervals with the gridpoints x₀ = 34 , x₁ = 40 , x₂ = 46 , x₃ = 52 , x₄ = 58 , and x₅ = 64. Use left and right Riemann sums to estimate how far Felix fell while traveling at supersonic speed.

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Properties of integrals Use only the fact that ∫₀⁴ 3𝓍 (4 ―𝓍) d𝓍 = 32, and the definitions and properties of integrals, to evaluate the following integrals, if possible.

(b) ∫₀⁴ 𝓍(𝓍 ― 4) d(𝓍)

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