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Ch. 5 - Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 5 - IntegrationProblema 5.4.31
Capitolo 5, Problema 5.4.31

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) = 𝓍ⁿ on [0,1] , for any positive integer n

Guida verificata passo dopo passo
1
Step 1: Recall the formula for the average value of a function ƒ(𝓍) on an interval [a, b]. The average value is given by: 1(b-a)∫fxdx. In this case, a = 0 and b = 1.
Step 2: Substitute ƒ(𝓍) = 𝓍ⁿ into the formula. The integral becomes: 11∫xndx, where the limits of integration are from 0 to 1.
Step 3: Compute the integral of 𝓍ⁿ. Use the power rule for integration: ∫xndx=xn+1n+1. Apply this rule to the integral.
Step 4: Evaluate the definite integral by substituting the limits of integration (0 and 1) into the result from Step 3. This gives: 1n+1[1n+1n+1-0n+1n+1].
Step 5: The average value of the function is the result of the evaluation in Step 4. To complete the problem, draw the graph of ƒ(𝓍) = 𝓍ⁿ on the interval [0, 1] and mark the average value as a horizontal line across the graph.

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Average Value of a Function

The average value of a function over a given interval is calculated using the formula (1/(b-a)) * ∫[a to b] f(x) dx, where [a, b] is the interval. This concept helps in understanding how the function behaves on average across the specified range, providing insight into its overall trend rather than just its individual values.
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Average Value of a Function

Definite Integral

A definite integral represents the accumulation of quantities, such as area under a curve, over a specific interval. It is denoted as ∫[a to b] f(x) dx and is fundamental in calculating the average value of a function, as it quantifies the total output of the function across the interval [a, b].
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Definition of the Definite Integral

Graphing Functions

Graphing a function involves plotting its values on a coordinate system, which visually represents its behavior. For the function f(x) = x^n, where n is a positive integer, the graph will show a curve that starts at (0,0) and rises to (1,1) as n increases, illustrating how the function's average value can be interpreted visually in relation to its shape.
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Graph of Sine and Cosine Function
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Domanda del libro di testo

Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.                                                                                  

                                                                                                                                                                    

 ∫ [(√𝓍 + 1)⁴ / 2√𝓍 d𝓍

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Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.                                                                                  

                                                                                                                                                                    

 ∫ (𝓍⁶ ― 3𝓍²)⁴ (𝓍⁵ ― 𝓍) d𝓍

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Use geometry and properties of integrals to evaluate


∫₀¹ (2𝓍 + √(1―𝓍²) + 1) d𝓍

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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) = 𝓍³ on [―1, 1]

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Area versus net area Graph the following functions. Then use geometry (not Riemann sums) to find the area and the net area of the region described.

The region between the graph of y = 1 - |x| and the x-axis, for -2 ≤ x ≤ 2

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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.


∫ₐ⁰ ƒ(𝓍) d𝓍

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