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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 5.2.53c
Chapter 5, Problem 5.2.53c

Properties of integrals Suppose βˆ«β‚€Β³Ζ’(𝓍) d𝓍 = 2 , βˆ«β‚ƒβΆΖ’(𝓍) d𝓍 = ―5 , and βˆ«β‚ƒβΆg(𝓍) d𝓍 = 1. Evaluate the following integrals.
(c) βˆ«β‚ƒβΆ (3Ζ’(𝓍) ― g(𝓍)) d𝓍

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Step 1: Recognize that the integral βˆ«β‚ƒβΆ (3Ζ’(𝓍) ― g(𝓍)) d𝓍 can be split into separate integrals using the linearity property of integrals. This property states that βˆ«β‚α΅‡ [c₁f(𝓍) + cβ‚‚g(𝓍)] d𝓍 = cβ‚βˆ«β‚α΅‡ f(𝓍) d𝓍 + cβ‚‚βˆ«β‚α΅‡ g(𝓍) d𝓍.
Step 2: Apply the linearity property to rewrite the integral as βˆ«β‚ƒβΆ (3Ζ’(𝓍)) d𝓍 ― βˆ«β‚ƒβΆ g(𝓍) d𝓍.
Step 3: Factor out the constant 3 from the first integral using the constant multiple rule, which states that βˆ«β‚α΅‡ cΒ·f(𝓍) d𝓍 = cΒ·βˆ«β‚α΅‡ f(𝓍) d𝓍. This gives 3βˆ«β‚ƒβΆ Ζ’(𝓍) d𝓍 ― βˆ«β‚ƒβΆ g(𝓍) d𝓍.
Step 4: Substitute the given values for the integrals. From the problem, βˆ«β‚ƒβΆ Ζ’(𝓍) d𝓍 = ―5 and βˆ«β‚ƒβΆ g(𝓍) d𝓍 = 1.
Step 5: Combine the results algebraically to evaluate the expression. The final result will be 3(―5) ― 1.

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Key Concepts

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

Properties of Definite Integrals

Definite integrals have several key properties, including linearity, which states that the integral of a sum is the sum of the integrals, and the integral of a constant multiplied by a function is the constant multiplied by the integral of the function. This allows us to break down complex integrals into simpler parts, making calculations more manageable.
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Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus connects differentiation and integration, stating that if a function is continuous on an interval, then the integral of its derivative over that interval equals the change in the function's values at the endpoints. This theorem is essential for evaluating definite integrals and understanding the relationship between a function and its integral.
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When dealing with piecewise functions, integrals can be evaluated over different intervals where the function may change. This requires calculating the integral separately over each interval and then summing the results. Understanding how to handle piecewise functions is crucial for accurately evaluating integrals that involve different expressions across specified ranges.
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Related Practice
Textbook Question

Sigma notation Evaluate the following expressions.                                                                                                                                          

(c)     4                                                                                                                                                                               

       βˆ‘ ΞΊΒ²                                                                                                                                                                          

       ΞΊ=1                         

Textbook Question

Zero net area Consider the function Ζ’(𝓍) = 𝓍² ― 4𝓍 .                                                                                                                                       

                                                                                                                                                                                     c) In general, for the function Ζ’(𝓍) = 𝓍² ― a𝓍, where a > 0, for what value of b > 0 (as a function of a) is βˆ«β‚€α΅‡ Ζ’(𝓍) d𝓍 = 0 ? 

Textbook Question

Matching functions with area functions Match the functions Ζ’, whose graphs are given in a― d, with the area functions A (𝓍) = βˆ«β‚€Λ£ Ζ’(t) dt, whose graphs are given in A–D.



Textbook Question

{Use of Tech} Approximating definite integrals Complete the following steps for the given integral and the given value of n. 

(c) Calculate the left and right Riemann sums for the given value of n.


βˆ«β‚€Β² (𝓍²―2) d𝓍 ; n = 4

Textbook Question

Approximating areas Estimate the area of the region bounded by the graph of Ζ’(𝓍) = xΒ² + 2 and the x-axis on [0, 2] in the following ways.

(c) Divide [0, 2] into n = 4 subintervals and approximate the area of the region using a right Riemann sum. Illustrate the solution geometrically.

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Textbook Question

{Use of Tech} Approximating definite integrals Complete the following steps for the given integral and the given value of n. 

(c) Calculate the left and right Riemann sums for the given value of n.


βˆ«β‚€^Ο€/2 cos 𝓍 d𝓍 ; n = 4

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