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Ch. 5 - Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
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Tutti i libri di testoBriggs 3rd EditionCh. 5 - IntegrationProblema 5.R.39
Capitolo 5, Problema 5.R.39

Evaluating integrals Evaluate the following integrals.


∫₋₂² (3𝓍⁴―2𝓍 + 1) d𝓍

Guida verificata passo dopo passo
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Step 1: Recognize that the integral ∫₋₂² (3𝓍⁴―2𝓍 + 1) d𝓍 is a definite integral, meaning we will evaluate the antiderivative of the function and then compute the difference between its values at the upper and lower limits.
Step 2: Break the integral into separate terms for easier computation: ∫₋₂² (3𝓍⁴) d𝓍 - ∫₋₂² (2𝓍) d𝓍 + ∫₋₂² (1) d𝓍.
Step 3: Compute the antiderivative of each term: For 3𝓍⁴, the antiderivative is (3/5)𝓍⁵; for -2𝓍, the antiderivative is -𝓍²; and for 1, the antiderivative is 𝓍.
Step 4: Apply the Fundamental Theorem of Calculus: Substitute the upper limit (𝓍 = 2) and lower limit (𝓍 = -2) into the antiderivative of each term, and compute the difference between the values at these limits.
Step 5: Combine the results from each term to find the total value of the definite integral. This involves adding the contributions from (3/5)𝓍⁵, -𝓍², and 𝓍 after evaluating them at the limits.

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Definite Integrals

A definite integral calculates the accumulation of a function's values over a specific interval, represented as ∫[a,b] f(x) dx. The result is a numerical value that represents the area under the curve of the function f(x) from x = a to x = b. Understanding the limits of integration and how they affect the area calculation is crucial for evaluating definite integrals.
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Definition of the Definite Integral

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus links differentiation and integration, stating that if F is an antiderivative of f on an interval [a, b], then ∫[a,b] f(x) dx = F(b) - F(a). This theorem allows us to evaluate definite integrals by finding the antiderivative of the integrand, simplifying the process of calculating areas under curves.
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Fundamental Theorem of Calculus Part 1

Polynomial Functions

Polynomial functions are expressions of the form f(x) = a_n x^n + a_(n-1) x^(n-1) + ... + a_1 x + a_0, where a_n are coefficients and n is a non-negative integer. In the given integral, the function 3x⁴ - 2x + 1 is a polynomial, and understanding how to integrate polynomial functions is essential, as they can be integrated term by term using the power rule.
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Taylor Polynomials
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(b) Find the average value of ƒ shown in the figure on the interval [2,6] and then find the point(s) c in (2, 6) guaranteed to exist by the Mean Value Theorem for Integrals. 

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Geometry of integrals Without evaluating the integrals, explain why the following statement is true for positive integers n:

∫₀¹ 𝓍ⁿd𝓍 + ∫₀¹ ⁿ√(𝓍d𝓍) = 1

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Area of regions Compute the area of the region bounded by the graph of ƒ and the 𝓍-axis on the given interval. You may find it useful to sketch the region.                                              

                                                                                                                                                                                    

 ƒ(𝓍) = 2 sin 𝓍/4 on [0, 2π]

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Properties of integrals Suppose ∫₁⁴ ƒ(𝓍) d𝓍 = 6 , ∫₁⁴ g(𝓍) d𝓍 = 4 and ∫₃⁴ ƒ(𝓍) d𝓍 = 2 . Evaluate the following integrals or state that there is not enough information.


∫₁³ ƒ(𝓍)/g(𝓍) d𝓍

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Area by geometry Use geometry to evaluate the following definite integrals, where the graph of ƒ is given in the figure.

(c) ∫₅⁷ ƒ(𝓍) d𝓍

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Evaluating integrals Evaluate the following integrals.


∫₁ᵉ d𝓍 / [𝓍(1 + ln 𝓍)]

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