교과서 질문
Why can the constant of integration be omitted from the antiderivative when evaluating a definite integral?
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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 5.2.63
Definite integrals from graphs The figure shows the areas of regions bounded by the graph of ƒ and the 𝓍-axis. Evaluate the following integrals.
∫₀ᶜ |ƒ(𝓍)| d𝓍
검증된 단계별 안내1
Identify the interval from 0 to c on the x-axis and observe the graph of the function ƒ(𝓍) over this interval.
Since the integral is of the absolute value |ƒ(𝓍)|, note that all areas between the graph and the x-axis will be considered positive, regardless of whether ƒ(𝓍) is above or below the x-axis.
Break the interval [0, c] into subintervals where the function ƒ(𝓍) does not change sign (i.e., where it is either entirely positive or entirely negative).
For each subinterval, calculate the area between the graph of ƒ(𝓍) and the x-axis. If the function is negative on that subinterval, take the absolute value of the integral (which corresponds to the area).
Sum all these positive areas from each subinterval to find the value of the definite integral ∫₀ᶜ |ƒ(𝓍)| d𝓍.
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Definite Integral
A definite integral calculates the net area under a curve between two points on the x-axis. It sums the values of the function over the interval, considering areas above the x-axis as positive and below as negative. This concept is fundamental for interpreting integrals from graphs.
추천 영상:
가이드 코스
05:43
Definition of the Definite Integral
Absolute Value of a Function in Integration
Integrating the absolute value of a function means summing all areas as positive, regardless of whether the function is above or below the x-axis. This ensures the integral represents total area without cancellation from negative parts, which is crucial when evaluating ∫₀ᶜ |ƒ(x)| dx.
추천 영상:
가이드 코스
06:37Average Value of a Function
Area Interpretation from Graphs
When given a graph, the definite integral corresponds to the area between the curve and the x-axis. Understanding how to read and calculate these areas, especially when the function crosses the axis, helps in evaluating integrals accurately, including those involving absolute values.
추천 영상:
가이드 코스
06:15Graphing The Derivative
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교과서 질문
Integrals with sin² 𝓍 and cos² 𝓍 Evaluate the following integrals.
∫ 𝓍 cos²𝓍² d𝓍
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교과서 질문
Symmetry of composite functions Prove that the integrand is either even or odd. Then give the value of the integral or show how it can be simplified. Assume f and g are even functions and p and q are odd functions.
∫ᵃ₋ₐ ƒ(g(𝓍)) d𝓍
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교과서 질문
Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.
∫ (sin⁵ 𝓍 + 3 sin³ 𝓍― sin 𝓍) cos 𝓍 d𝓍
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교과서 질문
Symmetry of composite functions Prove that the integrand is either even or odd. Then give the value of the integral or show how it can be simplified. Assume f and g are even functions and p and q are odd functions.
∫ᵃ₋ₐ ƒ(p(𝓍)) d𝓍
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교과서 질문
Suppose the interval [1, 3] is partitioned into n = 4 subintervals. What is the subinterval length ∆𝓍? List the grid points x₀ , x₁ , x₂ , x₃ and x₄. Which points are used for the left, right, and midpoint Riemann sums?
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