Ch. 8 - Techniques of Integration

Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo

Hass 15th Edition

Ch. 8 - Techniques of Integration

Problema 8.7.14a

Capitolo 8, Problema 8.7.14a

In Exercises 11–22, estimate the minimum number of subintervals needed to approximate the integrals with an error of magnitude less than 10^-4 by (a) the Trapezoidal Rule (The integrals in Exercises 11–18 are the integrals from Exercises 1–8.)
∫ from -2 to 0 of (x² - 1) dx

Guida verificata passo dopo passo

Identify the function to be integrated: $f(x) = x^{2} - 1$ over the interval $[-2, 0]$.

Recall the error bound formula for the Trapezoidal Rule: $|E_{T}| \leq \frac{(b - a)^{3}}{12 n^{2}} \max_{a \leq x \leq b} |f''(x)|$, where $n$ is the number of subintervals.

Compute the second derivative of the function: $f''(x) = \frac{d^{2}}{dx^{2}}(x^{2} - 1) = 2$.

Determine the maximum absolute value of $f''(x)$ on the interval $[-2, 0]$: since $f''(x) = 2$ is constant, $\max |f''(x)| = 2$.

Set the error bound less than the desired tolerance $10^{-4}$ and solve for $n$: \[ \frac{(0 - (-2))^{3}}{12 n^{2}} \times 2 < 10^{-4} \implies \frac{8}{12 n^{2}} \times 2 < 10^{-4} \implies \frac{16}{12 n^{2}} < 10^{-4} \]. Then solve this inequality for $n$ to find the minimum number of subintervals needed.

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Trapezoidal Rule

The Trapezoidal Rule is a numerical method to approximate definite integrals by dividing the interval into subintervals and approximating the area under the curve as trapezoids. The sum of these trapezoidal areas estimates the integral, with accuracy improving as the number of subintervals increases.

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Power Rules

Error Bound for the Trapezoidal Rule

The error bound for the Trapezoidal Rule estimates the maximum possible error in the approximation. It depends on the length of the interval, the number of subintervals, and the maximum value of the second derivative of the function on the interval. This bound helps determine how many subintervals are needed to achieve a desired accuracy.

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04:57

Determining Error and Relative Error

Second Derivative and Its Role in Error Estimation

The second derivative of the function measures its concavity and affects the error in the Trapezoidal Rule. The maximum absolute value of the second derivative on the interval is used in the error bound formula, indicating how 'curved' the function is and influencing the number of subintervals required for a given error tolerance.

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The Second Derivative Test: Finding Local Extrema

Pratica correlata

Domanda del libro di testo

88. The region in Exercise 87 is revolved about the x-axis to generate a solid.

a. Find the volume of the solid.

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Finding volume: Find the volume of the solid generated by revolving the region in the first quadrant bounded by the coordinate axes, the curve y = e^(-x), and the line x = 1.

a. About the y-axis.

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81. Find the values of p for which each integral converges.

a. ∫ from 1 to 2 of [dx / (x (ln x)^p)]

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In Exercises 11–22, estimate the minimum number of subintervals needed to approximate the integrals with an error of magnitude less than 10^-4 by (a) the Trapezoidal Rule (The integrals in Exercises 11–18 are the integrals from Exercises 1–8.)

∫ from -1 to 1 of (t³ + 1) dt

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∫ from 1 to 2 of 1/s² ds

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Finding area

Find the area of the region enclosed by the curve y = x cos(x) and the x-axis (see the accompanying figure) for:

a. π/2 ≤ x ≤ 3π/2.

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