교과서 질문
Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus
∫₁⁸ 8𝓍¹/³ d𝓍
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Ch. 5 - Integration
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
5장, 문제 5.4.43
Mean Value Theorem for Integrals Find or approximate all points at which the given function equals its average value on the given interval.
ƒ(𝓍) = 1 ― |𝓍| on [―1, 1]
검증된 단계별 안내1
Step 1: Recall the formula for the average value of a function on an interval [a, b]. The average value is given by: . For this problem, the interval is [−1, 1] and ƒ(𝓍) = 1 − |𝓍|.
Step 2: Compute the definite integral of ƒ(𝓍) over [−1, 1]. Break the integral into two parts because the absolute value function |𝓍| behaves differently for 𝓍 < 0 and 𝓍 ≥ 0. Specifically: = + .
Step 3: Simplify and evaluate each integral. For the first integral, , rewrite the integrand as . For the second integral, , the integrand remains . Compute these integrals separately.
Step 4: Divide the result of the definite integral by the length of the interval (b − a = 1 − (−1) = 2) to find the average value of the function on [−1, 1]. This gives the average value of ƒ(𝓍).
Step 5: Solve the equation ƒ(𝓍) = average value to find the points where the function equals its average value. Substitute ƒ(𝓍) = 1 − |𝓍| into the equation and solve for 𝓍. Consider both cases for |𝓍|: when 𝓍 ≥ 0 and when 𝓍 < 0.
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주요 개념
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Mean Value Theorem for Integrals
The Mean Value Theorem for Integrals states that if a function is continuous on a closed interval [a, b], then there exists at least one point c in (a, b) such that the value of the function at c equals the average value of the function over that interval. This theorem is essential for finding points where a function meets its average value.
추천 영상:
06:11
Fundamental Theorem of Calculus Part 1
Average Value of a Function
The average value of a continuous function f(x) over the interval [a, b] is calculated using the formula (1/(b-a)) * ∫[a to b] f(x) dx. This concept is crucial for determining the specific value that the function must equal at some point within the interval to satisfy the Mean Value Theorem for Integrals.
추천 영상:
06:37Average Value of a Function
Continuous Functions
A function is continuous on an interval if there are no breaks, jumps, or holes in its graph over that interval. Continuity is a key requirement for applying the Mean Value Theorem for Integrals, as it ensures that the function behaves predictably and that the average value can be accurately determined.
추천 영상:
05:34Intro to Continuity
관련 실천
교과서 질문
Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals.
∫₀^π/⁴ eˢᶦⁿ² ˣ sin 2𝓍 d𝓍
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교과서 질문
Identifying definite integrals as limits of sums Consider the following limits of Riemann sums for a function ƒ on [a,b]. Identify ƒ and express the limit as a definite integral.
n
lim ∑ (𝓍ₖ*² + 1) ∆𝓍ₖ on [0,2]
∆ → 0 k=1
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교과서 질문
If ƒ is an odd function, why is ∫ᵃ₋ₐ ƒ(𝓍) d𝓍 = 0?
169
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교과서 질문
Derivatives of integrals Simplify the following expressions.
d/d𝓍 ∫₃ˣ (t² + t + 1) dt
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교과서 질문
Definite integrals Use geometry (not Riemann sums) to evaluate the following definite integrals. Sketch a graph of the integrand, show the region in question, and interpret your result.
∫₋₁² ( ―|𝓍| ) d𝓍
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