Ch. 8 - Techniques of Integration

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 8 - Techniques of Integration

Problem 8.7.15a

Chapter 8, Problem 8.7.15a

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 0 to 2 of (t³ + t) dt

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Identify the function to be integrated: \(f(t) = t^{3} + t\) over the interval \([0, 2]\).

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

Compute the second derivative of the function: \(f''(t) = \frac{d^{2}}{dt^{2}}(t^{3} + t) = 6t\).

Find the maximum value of \(|f''(t)|\) on the interval \([0, 2]\): since \(f''(t) = 6t\) is increasing, the maximum is at \(t=2\), so \(\max |f''(t)| = 12\).

Set up the inequality for the error bound to be less than \(10^{-4}\) and solve for \(n\): \(\frac{(2 - 0)^{3}}{12 n^{2}} \times 12 < 10^{-4}\). Simplify and solve for \(n\) to find the minimum number of subintervals needed.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

Error Bound for the Trapezoidal Rule

The error bound for the Trapezoidal Rule depends on the second derivative of the function being integrated. Specifically, the error magnitude is at most (K(b - a)^3) / (12n^2), where K is the maximum absolute value of the second derivative on [a, b], and n is the number of subintervals. This formula helps determine how many subintervals are needed to achieve a desired accuracy.

Second Derivative and Its Role in Error Estimation

The second derivative of the integrand measures the function's concavity and affects the accuracy of the Trapezoidal Rule. A larger maximum second derivative on the interval implies a larger potential error, so calculating or estimating this value is essential for applying the error bound and deciding the number of subintervals.

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Related Practice

Textbook Question

Lifetime of a tire Assume the random variable L in Example 2f is normally distributed with mean μ = 22,000 miles and σ = 4,000 miles.

a. In a batch of 4000 tires, how many can be expected to last for at least 18,000 miles?

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Textbook Question

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 3 of (2x - 1) dx

Textbook Question

Consider the region bounded by the graphs of

y = ln(x), y = 0, and x = e.

a. Find the area of the region.

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Textbook Question

∫ from -1 to 1 of (t³ + 1) dt

Textbook Question

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.