Textbook Question
60. Two Methods
a. Evaluate ∫(x · ln(x²)) dx using the substitution u = x² and evaluating ∫(ln(u)) du.
Ch. 8 - Integration Techniques
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 8, Problem 8.8.68a
66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in the following exercises.
68. Let f(x) = e^(x²).
a. Find a Trapezoid Rule approximation to ∫[0 to 1] e^(x²) dx using n = 50 subintervals.
Verified step by step guidance1
Step 1: Recall the Trapezoid Rule formula for approximating definite integrals: \( T_n = \frac{\Delta x}{2} \left[ f(x_0) + 2f(x_1) + 2f(x_2) + \dots + 2f(x_{n-1}) + f(x_n) \right] \), where \( \Delta x = \frac{b-a}{n} \). Here, \( a = 0 \), \( b = 1 \), and \( n = 50 \).
Step 2: Calculate \( \Delta x \), the width of each subinterval: \( \Delta x = \frac{1 - 0}{50} = \frac{1}{50} \). This gives the spacing between the points \( x_0, x_1, \dots, x_{50} \).
Step 3: Determine the points \( x_i \) for \( i = 0, 1, \dots, 50 \) using \( x_i = a + i \Delta x \). For example, \( x_0 = 0 \), \( x_1 = \frac{1}{50} \), \( x_2 = \frac{2}{50} \), and so on up to \( x_{50} = 1 \).
Step 4: Evaluate \( f(x_i) = e^{x_i^2} \) at each of the points \( x_0, x_1, \dots, x_{50} \). For example, \( f(x_0) = e^{0^2} = e^0 = 1 \), \( f(x_1) = e^{(\frac{1}{50})^2} \), \( f(x_2) = e^{(\frac{2}{50})^2} \), and so on.
Step 5: Substitute the values of \( f(x_i) \) and \( \Delta x \) into the Trapezoid Rule formula to compute the approximation. Remember to multiply the first and last terms by 1 and all intermediate terms by 2, then sum them up and multiply by \( \frac{\Delta x}{2} \).
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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 integration technique that approximates the area under a curve by dividing it into trapezoids rather than rectangles. For a function f(x) over an interval [a, b], the area is estimated by calculating the average of the function values at the endpoints, multiplied by the width of the interval. This method improves accuracy compared to using rectangles, especially when the function is relatively linear over small subintervals.
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Power Rules
Numerical Integration
Numerical integration refers to a set of algorithms for calculating the integral of a function when an analytical solution is difficult or impossible to obtain. Techniques such as the Trapezoidal Rule and Simpson's Rule are commonly used to estimate definite integrals by approximating the area under the curve. These methods are particularly useful in applied mathematics, physics, and engineering, where exact solutions may not be feasible.
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Error Estimation
Error estimation in numerical integration involves determining the difference between the exact value of an integral and its numerical approximation. Theorem 8.1 likely provides a framework for estimating this error, which can depend on factors such as the number of subintervals used and the behavior of the function being integrated. Understanding error estimation is crucial for assessing the reliability of numerical results and ensuring that they meet the required precision.
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Related Practice
Textbook Question
57. Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. If x = 4 tanθ, then cscθ = 4/x.
Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. Suppose ∫_a^b f(x) dx is approximated with Simpson’s Rule using n = 18 subintervals, where |f^(4)(x)| ≤ 1 on [a, b]. The absolute error E_S in approximating the integral satisfies E_S ≤ (Δx)^5 / 10.
Textbook Question
66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in the following exercises.
66. Let f(x) = cos(x²).
a. Find a Midpoint Rule approximation to ∫[-1 to 1] cos(x²) dx using n = 30 subintervals.
Textbook Question
62. Electronic Chips Suppose the probability that a particular computer chip fails after a hours of operation is 0.00005 ∫(from a to ∞) e^(-0.00005t) dt.
a. Find the probability that the computer chip fails after 15,000 hr of operation.
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Textbook Question
66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in the following exercises.
71. Let f(x) = √(sin x).
a. Find a Simpson's Rule approximation to the integral from 1 to 2 of √(sin x) dx using n = 20 subintervals.