Domanda del libro di testo
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.
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Ch. 8 - Integration Techniques
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- 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: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 8, Problema 8.9.62a
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.
Guida verificata passo dopo passo1
Identify the given probability expression: the probability that the chip fails after \( a \) hours is given by \( 0.00005 \int_{a}^{\infty} e^{-0.00005t} \, dt \).
To find the probability that the chip fails after 15,000 hours, substitute \( a = 15000 \) into the integral: \( 0.00005 \int_{15000}^{\infty} e^{-0.00005t} \, dt \).
Evaluate the improper integral \( \int_{15000}^{\infty} e^{-0.00005t} \, dt \). Recall that the integral of \( e^{kt} \) with respect to \( t \) is \( \frac{1}{k} e^{kt} \), where \( k \) is a constant.
Apply the limits of integration from 15,000 to infinity. Since \( e^{-0.00005t} \) approaches zero as \( t \to \infty \), the upper limit contributes zero to the definite integral.
Multiply the result of the integral by 0.00005 to find the final probability expression for the chip failing after 15,000 hours.
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Improper Integrals
Improper integrals involve integration over an infinite interval or integrands with infinite discontinuities. In this problem, the integral from a finite value to infinity calculates the probability that the chip lasts beyond a certain time, requiring evaluation of an integral with an infinite upper limit.
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Percorso guidato
11:11
Improper Integrals: Infinite Intervals
Exponential Decay Function
The function e^(-kt) models exponential decay, describing processes where quantities decrease at rates proportional to their current value. Here, it represents the decreasing probability density of chip failure over time, with the parameter 0.00005 controlling the decay rate.
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09:29Exponential Growth & Decay
Probability and Cumulative Distribution Function (CDF)
The integral of a probability density function (PDF) from a point to infinity gives the probability that a random variable exceeds that point. This is related to the survival function or complementary CDF, which in this case gives the probability the chip operates beyond a specified time.
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3:56Slope-Intercept Form
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Domanda del libro di testo
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.
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Domanda del libro di testo
2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
29. ∫ cos⁴ x/sin⁶ x dx
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Domanda del libro di testo
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.
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Domanda del libro di testo
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.
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Domanda del libro di testo
2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
48. ∫ sin(3x) cos⁶(3x) dx
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