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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.R.39
Chapter 5, Problem 5.R.39

Evaluating integrals Evaluate the following integrals.


∫₋₂² (3𝓍⁴―2𝓍 + 1) d𝓍

Verified step by step guidance
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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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Key Concepts

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

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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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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Related Practice
Textbook Question

Find the intervals on which ƒ(𝓍) = ∫ₓ¹ (t―3) (t―6)¹¹ dt is increasing and the intervals on which it is decreasing.

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Textbook Question

Evaluate the following derivatives.


d/d𝓍 ∫₃ᵉˣ cos t² dt

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Textbook Question

Evaluating integrals Evaluate the following integrals.


∫π/₆^π/³ (sec² t + csc² t) dt

Textbook Question

Area functions and the Fundamental Theorem Consider the function

ƒ(t) = { t      if  ―2 ≤ t < 0

t²/2    if    0 ≤ t ≤ 2

and its graph shown below. Let F(𝓍) = ∫₋₁ˣ ƒ(t) dt and G(𝓍) = ∫₋₂ˣ ƒ(t) dt.

(b) Use the Fundamental Theorem to find an expression for F '(𝓍) for ―2 ≤ 𝓍 < 0.

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Textbook Question

Area functions and the Fundamental Theorem Consider the function

ƒ(t) = { t      if  ―2 ≤ t < 0

t²/2    if    0 ≤ t ≤ 2

and its graph shown below. Let F(𝓍) = ∫₋₁ˣ ƒ(t) dt and G(𝓍) = ∫₋₂ˣ ƒ(t) dt.

(e) Evaluate F ''(―1) and F ''(1). Interpret these values.

Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume ƒ and ƒ' are continuous functions for all real numbers.

(d) If ƒ is continuous on [a,b] and ∫ₐᵇ |ƒ(𝓍)| d𝓍 = 0 , then ƒ(𝓍) = 0 on [a,b] .

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