Textbook Question
In Exercises 69–80, determine whether the improper integral converges or diverges. If it converges, evaluate the integral.
∫₋∞⁰ x² e^(x³) dx
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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 Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 8, Problem 8.7.32
The length of one arch of the curve y = sin x is given by
L = ∫(from 0 to π) √(1 + cos²(x)) dx.
Estimate L by Simpson's Rule with n = 8.
Verified step by step guidance1
Identify the function to integrate: here, the integrand is \( f(x) = \sqrt{1 + \cos^{2}(x)} \).
Determine the interval and number of subintervals: the integral is from \( 0 \) to \( \pi \), and \( n = 8 \) subintervals means the step size is \( h = \frac{\pi - 0}{8} = \frac{\pi}{8} \).
Calculate the values of \( f(x) \) at the points \( x_0 = 0, x_1 = h, x_2 = 2h, \ldots, x_8 = 8h = \pi \). For each \( x_i \), compute \( f(x_i) = \sqrt{1 + \cos^{2}(x_i)} \).
Apply Simpson's Rule formula: \[ L \approx \frac{h}{3} \left[ f(x_0) + 4 \sum_{i=1,3,5,7} f(x_i) + 2 \sum_{i=2,4,6} f(x_i) + f(x_8) \right] \].
Sum the weighted function values according to Simpson's Rule and multiply by \( \frac{h}{3} \) to estimate the length \( L \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Arc Length of a Curve
The arc length of a curve y = f(x) from x = a to x = b is given by the integral L = ∫_a^b √(1 + (dy/dx)²) dx. This formula measures the distance along the curve by summing infinitesimal line segments, accounting for changes in both x and y.
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Arc Length of Parametric Curves
Simpson's Rule
Simpson's Rule is a numerical method to approximate definite integrals by dividing the interval into an even number of subintervals and fitting parabolas through the points. It provides an accurate estimate by weighting function values at endpoints, midpoints, and interior points.
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5:50Power Rules
Partitioning the Interval (n = 8)
Choosing n = 8 means dividing the interval [0, π] into 8 equal subintervals for applying Simpson's Rule. This partitioning determines the step size and the points at which the integrand is evaluated, affecting the accuracy of the numerical approximation.
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Related Practice
Textbook Question
Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.
∫ √(x² - 4) / x dx
Textbook Question
The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.
∫ (dx / ((x - 2)√(x² - 4x + 3)))
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Textbook Question
In Exercises 39–48, use an appropriate substitution and then a trigonometric substitution to evaluate the integrals.
∫ (e^{t} dt) / ((1 + e^{2t})^{3/2}) from ln(3/4) to ln(4/3)
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Textbook Question
Solve the initial value problems in Exercises 53–56 for y as a function of x.
(x² + 1)² (dy/dx) = √(x² + 1), where y(0) = 1
Textbook Question
Use reduction formulas to evaluate the integrals in Exercises 41–50.
∫ 8 cot^4(t) dt