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Ch. 5 - Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 5 - IntegrationProblem 5.2.39
Chapter 5, Problem 5.2.39

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.                                                                                                                                      
                                                                                                                                                                                       
 βˆ«β‚€β΄ (8―2𝓍) d𝓍

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Step 1: Recognize that the integrand, 8 - 2𝓍, represents a linear function. The graph of this function is a straight line with a y-intercept of 8 and a slope of -2. Sketch the graph of this line over the interval [0, 4].
Step 2: Identify the region enclosed by the graph of the function, the x-axis, and the vertical lines x = 0 and x = 4. This region forms a trapezoid.
Step 3: Break the trapezoid into simpler geometric shapes if necessary. In this case, the trapezoid can be analyzed directly using the formula for the area of a trapezoid: Area = (1/2) Γ— (Base₁ + Baseβ‚‚) Γ— Height.
Step 4: Calculate the dimensions of the trapezoid. Base₁ is the value of the function at x = 0, which is 8. Baseβ‚‚ is the value of the function at x = 4, which is 0. The height of the trapezoid is the distance between x = 0 and x = 4, which is 4.
Step 5: Substitute the dimensions into the formula for the area of a trapezoid to find the value of the definite integral. Interpret the result as the total area under the curve from x = 0 to x = 4.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Definite Integrals

A definite integral represents the signed area under a curve defined by a function over a specific interval [a, b]. It is calculated using the Fundamental Theorem of Calculus, which connects differentiation and integration. The result of a definite integral is a number that quantifies the total accumulation of the quantity represented by the function between the limits of integration.
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Geometric Interpretation

The geometric interpretation of definite integrals involves visualizing the area between the curve of the integrand and the x-axis over the specified interval. This area can be positive or negative depending on whether the curve lies above or below the x-axis. Understanding this concept allows for evaluating integrals using geometric shapes, such as rectangles, triangles, or trapezoids, rather than relying solely on algebraic methods.
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Sketching Graphs

Sketching the graph of the integrand is crucial for visualizing the function's behavior over the interval of integration. It helps identify key features such as intercepts, maxima, minima, and the overall shape of the curve. A well-drawn graph aids in understanding the area to be calculated and can simplify the evaluation of the definite integral by allowing for geometric reasoning.
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Related Practice
Textbook Question

Max/min of area functions Suppose Ζ’ is continuous on [0 ,∞) and A(𝓍) is the net area of the region bounded by the graph of Ζ’ and the t-axis on [0, x]. Show that the local maxima and minima of A occur at the zeros of Ζ’. Verify this fact with the function Ζ’(𝓍) = 𝓍² - 10𝓍.

Textbook Question

Variations on the substitution method Evaluate the following integrals.                                                                                                        

                                                                                                                                                                    

 βˆ« (eΛ£ ― e⁻ˣ)/ (eΛ£ + e⁻ˣ) d𝓍

Textbook Question

Explain the statement that a continuous function on an interval [a,b] equals its average value at some point on (a,b).

Textbook Question

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.

Ζ’(𝓍) = cos 𝓍 on [―π/2 , Ο€/2]

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Textbook Question

Areas of regions Find the area of the following regions.                                                                                                                   

                                                                                                                                                                 The region bounded by the graph of Ζ’(𝓍) = (𝓍―4)⁴ and the 𝓍-axis between and 𝓍 = 2 and π“= 6

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Textbook Question

Integrals with sinΒ² 𝓍 and cosΒ² 𝓍 Evaluate the following integrals.                                                                                                             

                                                                                                                                                                    

 βˆ«β‚€^Ο€/⁴ cosΒ² 8ΞΈ dΞΈ