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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 5.3.111c
Chapter 5, Problem 5.3.111c

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 ? 

Verified step by step guidance
1
Start by writing down the integral you need to solve: \(\int_0^b (x^2 - a x) \, dx = 0\), where \(a > 0\) and \(b > 0\).
Compute the indefinite integral of the function \(f(x) = x^2 - a x\). The antiderivative is \(\int (x^2 - a x) \, dx = \frac{x^3}{3} - \frac{a x^2}{2} + C\).
Evaluate the definite integral from 0 to \(b\) using the antiderivative: \(\left[ \frac{x^3}{3} - \frac{a x^2}{2} \right]_0^b = \frac{b^3}{3} - \frac{a b^2}{2} - \left(0\right)\).
Set the definite integral equal to zero to find \(b\): \(\frac{b^3}{3} - \frac{a b^2}{2} = 0\).
Factor the equation to solve for \(b\): \(b^2 \left( \frac{b}{3} - \frac{a}{2} \right) = 0\). Since \(b > 0\), solve \(\frac{b}{3} - \frac{a}{2} = 0\) for \(b\) to express \(b\) as a function of \(a\).

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Key Concepts

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

Definite Integral and Net Area

The definite integral of a function over an interval represents the net area between the function's graph and the x-axis. Positive areas above the x-axis add to the integral, while areas below subtract. When the integral equals zero, the positive and negative areas cancel out, resulting in zero net area.
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Definition of the Definite Integral

Finding the Integral of a Quadratic Function

To evaluate the integral of a quadratic function like Ζ’(x) = xΒ² - a x, you apply the power rule for integration term-by-term. This involves increasing the exponent by one and dividing by the new exponent, then applying limits to find the definite integral value as a function of the upper limit b.
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Integrals of General Exponential Functions

Solving for the Upper Limit to Achieve Zero Net Area

Setting the definite integral equal to zero and solving for the upper limit b involves forming an equation from the integral expression and isolating b. This process finds the point where the accumulated positive and negative areas balance, which depends on the parameter a in the function.
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Finding Area Between Curves on a Given Interval
Related Practice
Textbook Question

Sigma notation Evaluate the following expressions.                                                                                                                                          

(c)     4                                                                                                                                                                               

       βˆ‘ ΞΊΒ²                                                                                                                                                                          

       ΞΊ=1                         

Textbook Question

Properties of integrals Use only the fact that βˆ«β‚€β΄ 3𝓍 (4 ―𝓍) d𝓍 = 32, and the definitions and properties of integrals, to evaluate the following integrals, if possible.


(c) βˆ«β‚„β° 6𝓍(4 ― 𝓍) d(𝓍)

Textbook Question

Working with area functions Consider the function Ζ’ and its graph.

(c) Sketch a graph of A, for 0 ≀ 𝓍 ≀ 10 , without a scale on the y-axis.


Textbook Question

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

Textbook Question

Sigma notation Express the following sums using sigma notation. (Answers are not unique.)

(c) 1Β² + 2Β² + 3Β² + 4Β²

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