교과서 질문
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.
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Ch. 8 - Techniques of Integration
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- 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
8장, 문제 8.7.20a
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 0 to 3 of 1/√(x + 1) dx
검증된 단계별 안내1
Identify the function to be integrated: \(f(x) = \frac{1}{\sqrt{x + 1}}\) over the interval \([0, 3]\).
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 first derivative \(f'(x)\) and then the second derivative \(f''(x)\) of the function \(f(x)\).
Find the maximum absolute value of \(f''(x)\) on the interval \([0, 3]\) by analyzing \(f''(x)\) or evaluating it at critical points and endpoints.
Set the error bound \(\frac{(3 - 0)^3}{12 n^2} \max |f''(x)| < 10^{-4}\) and solve this inequality for \(n\) to estimate the minimum number of subintervals needed.
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주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
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.
추천 영상:
5:50
Power Rules
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 integrand and the number of subintervals. Specifically, the error is bounded by (b−a)^3/(12n^2) times the maximum of |f''(x)| on [a,b].
추천 영상:
가이드 코스
04:57Determining Error and Relative Error
Second Derivative and Its Role in Error Estimation
The second derivative of the function indicates the concavity and affects 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, which helps determine how many subintervals are needed for a desired accuracy.
추천 영상:
06:02The Second Derivative Test: Finding Local Extrema
관련 실천
교과서 질문
88. The region in Exercise 87 is revolved about the x-axis to generate a solid.
a. Find the volume of the solid.
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교과서 질문
Consider the region bounded by the graphs of
y = ln(x), y = 0, and x = e.
a. Find the area of the region.
32
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교과서 질문
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
30
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교과서 질문
Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
∫ (2 − cosx + sinx) / sin²x dx
30
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교과서 질문
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
30
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