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Ch. 5 - Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Briggs 3rd EditionCh. 5 - Integration문제 5.2.59
5장, 문제 5.2.59

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.




∫₀ᵃ ƒ(𝓍) d𝓍

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Identify the regions bounded by the graph of the function ƒ(𝓍) and the 𝓍-axis between 𝓍 = 0 and 𝓍 = a. Note whether these regions lie above or below the 𝓍-axis, as this affects the sign of the integral.
Recall that the definite integral ∫₀ᵃ ƒ(𝓍) d𝓍 represents the net area between the curve and the 𝓍-axis from 0 to a. Areas above the axis contribute positively, while areas below contribute negatively.
Determine the numerical values of the areas shown in the figure for each region between 0 and a. Assign positive values to areas above the axis and negative values to areas below the axis.
Sum all these signed areas to find the value of the definite integral ∫₀ᵃ ƒ(𝓍) d𝓍. This sum represents the net area under the curve from 0 to a.
Write the final expression for the integral as the sum of these areas, ensuring to include the correct signs based on the position relative to the 𝓍-axis.

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주요 개념

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Definite Integral as Net Area

A definite integral from a to b of a function f(x) represents the net area between the graph of f and the x-axis over [a, b]. Areas above the x-axis contribute positively, while areas below contribute negatively, resulting in the integral's value.
추천 영상:
가이드 코스
05:43
Definition of the Definite Integral

Interpreting Graphs for Integration

When evaluating integrals from graphs, it is essential to identify the regions bounded by the curve and the x-axis, noting their shapes and whether they lie above or below the axis. This helps in calculating areas accurately, often by summing or subtracting given region areas.
추천 영상:
가이드 코스
06:15
Graphing The Derivative

Properties of Definite Integrals

Definite integrals have properties such as additivity over intervals and sign changes when limits are reversed. Understanding these properties allows breaking complex integrals into simpler parts and correctly combining areas to find the total integral value.
추천 영상:
가이드 코스
05:43
Definition of the Definite Integral
관련 실천
교과서 질문

Max/min of area functions Suppose ƒ is continuous on [0 ,∞) and A(𝓍) is the net area of the region bounded by the graph of ƒ and the t-axis on [0, x]. Show that the local maxima and minima of A occur at the zeros of ƒ. Verify this fact with the function ƒ(𝓍) = 𝓍² - 10𝓍.

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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 (8―2𝓍) d𝓍

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교과서 질문

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교과서 질문

Average values Find the average value of the following functions on the given interval. Draw a graph of the function and indicate the average value.

ƒ(𝓍) = cos 𝓍 on [―π/2 , π/2]

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교과서 질문

If ƒ is an odd function, why is ∫ᵃ₋ₐ ƒ(𝓍) d𝓍 = 0?

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교과서 질문

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 ∫₀^π/⁴ cos² 8θ dθ

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