Textbook Question
4. What substitutions are made to evaluate integrals of sin(mx)sin(nx), sin(mx)cos(nx), and cos(mx)cos(nx)? Give an example of each case.
Ch. 8 - Techniques of Integration
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 8, Problem 8.7.17a
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 2 of 1/s² ds
Verified step by step guidance1
Identify the function to be integrated: \(f(x) = \frac{1}{x^2}\), and the interval of integration: \([1, 2]\).
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 \(f(x)\). Start by finding the first derivative: \(f'(x) = -2 x^{-3}\), then the second derivative: \(f''(x) = 6 x^{-4}\).
Determine the maximum value of \(|f''(x)|\) on the interval \([1, 2]\). Since \(f''(x) = 6 / x^4\) is positive and decreasing on \([1, 2]\), the maximum occurs at \(x = 1\), so \(\max |f''(x)| = 6\).
Set up the inequality for the error bound to be less than \(10^{-4}\): \(\frac{(2 - 1)^3}{12 n^2} \times 6 < 10^{-4}\). 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 accuracy depends on the number of subintervals; more subintervals generally yield better approximations.
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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 second derivative of the function, the length of the interval, and the number of subintervals. Specifically, the error is bounded by (b−a)³/(12n²) times the maximum of |f''(x)| on [a,b].
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04:57Determining Error and Relative Error
Second Derivative and Its Role in Error Estimation
The second derivative of the integrand measures the concavity of the function and influences the error in the Trapezoidal Rule. To estimate the error bound, one must find the maximum absolute value of the second derivative on the interval, as it directly affects how closely the trapezoids approximate the curve.
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06:02The Second Derivative Test: Finding Local Extrema
Related Practice
Textbook Question
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.
Textbook Question
Heat capacity of a gas
Heat capacity
C_v
is the amount of heat required to raise the temperature of a given mass of gas with constant volume by 1°C, measured in units of cal/deg-mol (calories per degree gram molecular weight).
The heat capacity of oxygen depends on its temperature T and satisfies the formula
C_v = 8.27 + 10^(-5) * (26T − 1.87T²)
Use Simpson’s Rule to find the average value of C_v and the temperature at which it is attained for
20°C ≤ T ≤ 675°C.
Textbook Question
Evaluate the integrals in Exercises 29–32 (b) using a trigonometric substitution.
∫ [x / √(4 − x²)] dx
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.
1
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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 1 of (x² + 1) dx