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Ch. 8 - Integration Techniques
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. 8 - Integration TechniquesProblema 8.8.6
Capitolo 8, Problema 8.8.6

5-8. Compute the following estimates of ∫(0 to 8) f(x) dx using the graph in the figure.
Graph of a function with labeled axes, showing a fluctuating curve plotted on a grid.
6. T(4)

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1
Identify that T(4) refers to the Trapezoidal Rule approximation of the integral using 4 subintervals over the interval [0, 8].
Calculate the width of each subinterval, \( \Delta x = \frac{8 - 0}{4} = 2 \).
Determine the x-values at the endpoints of the subintervals: 0, 2, 4, 6, and 8.
From the graph, find the corresponding function values \( f(x) \) at these points: \( f(0), f(2), f(4), f(6), f(8) \).
Apply the Trapezoidal Rule formula: \[ T(4) = \frac{\Delta x}{2} \left[ f(0) + 2f(2) + 2f(4) + 2f(6) + f(8) \right] \] This will give the estimate of the integral \( \int_0^8 f(x) \, dx \).

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Definite Integral as Area Under a Curve

The definite integral of a function over an interval represents the net area between the function's graph and the x-axis. It can be approximated by summing areas of shapes like rectangles or trapezoids under the curve, which is essential for estimating integrals from graphs.
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Definition of the Definite Integral

Trapezoidal Rule

The trapezoidal rule approximates the integral by dividing the interval into subintervals and approximating the area under the curve as trapezoids rather than rectangles. The formula averages the function values at the endpoints of each subinterval, providing a more accurate estimate than simple rectangular sums.
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Power Rules

Partitioning the Interval

To apply numerical integration methods like the trapezoidal rule, the interval of integration is divided into equal or specified subintervals. For T(4), the interval [0,8] is divided into 4 subintervals of length 2, and function values at these points are used to calculate the trapezoidal sum.
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Interval of Convergence
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11-14. {Use of Tech} Compute the absolute and relative errors in using c to approximate x.

12. x = √2; c = 1.414

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"Electric field due to a line of charge A total charge of Q is distributed uniformly on a line segment of length 2L along the y-axis (see figure). The x-component of the electric field at a point (a, 0) is given by

Eₓ(a) = (kQa/2L) ∫-L L dy/(a² + y²)^(3/2),

where k is a physical constant and a > 0.

a. Confirm that Eₓ(a)=kQ / a √(a²+L²)

b. Letting ρ=Q / 2 L be the charge density on the line segment, show that if L → ∞, then Eₓ(a) = 2kρ / a.

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60–69. Completing the square Evaluate the following integrals.

62. ∫ du / (2u² - 12u + 36)

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29-34. {Use of Tech} Comparing the Midpoint and Trapezoid Rules

Apply the Midpoint and Trapezoid Rules to the following integrals. Make a table similar to Table 8.5 showing the approximations and errors for n = 4, 8, 16, and 32. The exact values of the integrals are given for computing the error.

30. ∫(0 to 6) (x³/16 - x) dx = 4

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2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.

10. ∫ (x³ + 3x² + 1)/(x³ + 1) dx

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108. Arc length Find the length of the curve y = ln(x) on the interval [1, e^2].

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