Skip to main content
Ch. 8 - Integration Techniques
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Briggs 3rd EditionCh. 8 - Integration Techniques문제 8.8.71a
8장, 문제 8.8.71a

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.

검증된 단계별 안내
1
Step 1: Understand the problem. We are tasked with approximating the integral of f(x) = √(sin x) over the interval [1, 2] using Simpson's Rule with n = 20 subintervals. Simpson's Rule is a numerical integration method that uses parabolic approximations to estimate the area under a curve.
Step 2: Recall the formula for Simpson's Rule: \( S_n = \frac{\Delta x}{3} \left[ f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + \dots + 4f(x_{n-1}) + f(x_n) \right] \), where \( \Delta x = \frac{b-a}{n} \), \( x_0, x_1, \dots, x_n \) are the subinterval points, and \( n \) must be even.
Step 3: Calculate \( \Delta x \). Since \( a = 1 \), \( b = 2 \), and \( n = 20 \), \( \Delta x = \frac{2-1}{20} = 0.05 \). The subinterval points are \( x_0 = 1, x_1 = 1.05, x_2 = 1.1, \dots, x_{20} = 2 \).
Step 4: Evaluate \( f(x) = \sqrt{\sin x} \) at each subinterval point \( x_0, x_1, \dots, x_{20} \). For example, \( f(x_0) = \sqrt{\sin(1)} \), \( f(x_1) = \sqrt{\sin(1.05)} \), and so on. These values will be used in the Simpson's Rule formula.
Step 5: Apply the Simpson's Rule formula. Substitute \( \Delta x = 0.05 \), the calculated \( f(x_i) \) values, and the coefficients (1 for \( f(x_0) \) and \( f(x_n) \), 4 for odd-indexed terms, and 2 for even-indexed terms) into the formula. Simplify the expression to approximate the integral.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
7m
도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Simpson's Rule

Simpson's Rule is a numerical method for approximating the definite integral of a function. It uses parabolic segments to estimate the area under a curve, providing a more accurate approximation than methods like the Trapezoidal Rule. The formula involves evaluating the function at evenly spaced points and applying weights to these values, which is particularly useful for functions that are smooth and continuous.
추천 영상:
가이드 코스
5:50
Power Rules

Definite Integral

A definite integral represents the signed area under a curve defined by a function over a specific interval [a, b]. It is calculated using the Fundamental Theorem of Calculus, which connects differentiation and integration. The result of a definite integral is a number that quantifies the accumulation of quantities, such as area, over the interval.
추천 영상:
05:43
Definition of the Definite Integral

Error Estimation in Numerical Integration

Error estimation in numerical integration involves determining how closely a numerical approximation, like Simpson's Rule, approximates the actual value of the integral. Theorem 8.1 likely provides a formula or method to estimate this error based on the function's behavior and the number of subintervals used. Understanding error estimation is crucial for assessing the reliability of numerical results.
추천 영상:
04:57
Determining Error and Relative Error
관련 실천
교과서 질문

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.

48
views
교과서 질문

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

159
views
교과서 질문

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.

31
views
교과서 질문

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.

45
views
교과서 질문

2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.

51. ∫ (from 0 to π/4) sin⁵(4θ) dθ

62
views
교과서 질문

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

98
views