교과서 질문
Area Find (i) the net area and (ii) the area of the following regions. Graph the function and indicate the region in question.
The region bounded by y = 6 cos 𝓍 and the 𝓍-axis between 𝓍 = ―π/2 and 𝓍 = π
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Ch. 5 - Integration
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
5장, 문제 5.R.86
Evaluating integrals Evaluate the following integrals.
∫₀⁵ |2𝓍―8|d𝓍
검증된 단계별 안내1
Step 1: Recognize that the integral involves the absolute value function |2𝓍 - 8|. Absolute value functions can be split into piecewise functions based on where the expression inside the absolute value changes sign.
Step 2: Solve for the point where 2𝓍 - 8 = 0. This occurs when 𝓍 = 4. Therefore, the integral can be split into two intervals: [0, 4] and [4, 5].
Step 3: On the interval [0, 4], the expression 2𝓍 - 8 is negative, so |2𝓍 - 8| = -(2𝓍 - 8). On the interval [4, 5], the expression 2𝓍 - 8 is positive, so |2𝓍 - 8| = 2𝓍 - 8.
Step 4: Rewrite the integral as the sum of two integrals: ∫₀⁴ -(2𝓍 - 8)d𝓍 + ∫₄⁵ (2𝓍 - 8)d𝓍. Simplify the integrands in each interval.
Step 5: Compute each integral separately by applying the power rule for integration and evaluating the definite integrals. Combine the results to find the total value of the original integral.
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
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주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Definite Integrals
A definite integral calculates the accumulation of a quantity, represented as the area under a curve between two specified limits. In this case, the integral ∫₀⁵ |2𝓍―8|d𝓍 evaluates the area between the curve of the function |2𝓍―8| and the x-axis from x=0 to x=5. Understanding how to set up and evaluate definite integrals is crucial for solving problems involving areas and accumulated quantities.
추천 영상:
05:43
Definition of the Definite Integral
Absolute Value Function
The absolute value function, denoted as |x|, outputs the non-negative value of x regardless of its sign. In the integral ∫₀⁵ |2𝓍―8|d𝓍, the expression inside the absolute value, 2𝓍―8, can be positive or negative depending on the value of x. Identifying where the expression changes sign is essential for correctly evaluating the integral, as it affects the area calculation.
추천 영상:
06:37Average Value of a Function
Piecewise Functions
Piecewise functions are defined by different expressions based on the input value. For the integral involving |2𝓍―8|, we need to determine the points where 2𝓍―8 equals zero to split the integral into segments where the function behaves differently. This approach allows us to evaluate the integral accurately by considering the appropriate expression for each segment of the interval.
추천 영상:
05:36Piecewise Functions
관련 실천
교과서 질문
Evaluating integrals Evaluate the following integrals.
∫₁⁴ ((√v + v) / v ) dv
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교과서 질문
Consider the function
ƒ(t) = { t if ―2 ≤ t < 0
t²/2 if 0 ≤ t ≤ 2
and its graph shown below. Let F(𝓍) = ∫₋₁ˣ ƒ(t) dt and G(𝓍) = ∫₋₂ˣ ƒ(t) dt.
(f) Find a constant C such that F(𝓍) = G(𝓍) + C .
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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.
∫₀⁴ ƒ(𝓍) d𝓍, where ƒ(𝓍) = {5 if 𝓍 ≤ 2
3𝓍 ― 1 if 𝓍 > 2
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교과서 질문
Integration by Riemann sums Consider the integral ∫₁⁴ (3𝓍― 2) d𝓍.
(a) Evaluate the right Riemann sum for the integral with n = 3 .
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교과서 질문
Variations on the substitution method Evaluate the following integrals.
∫ (eˣ ― e⁻ˣ)/ (eˣ + e⁻ˣ) d𝓍
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