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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 5.3.111c
Capitolo 5, Problema 5.3.111c

Zero net area Consider the function ƒ(𝓍) = 𝓍² ― 4𝓍 .                                                                                                                                       
                                                                                                                                                                                     c) In general, for the function ƒ(𝓍) = 𝓍² ― a𝓍, where a > 0, for what value of b > 0 (as a function of a) is ∫₀ᵇ ƒ(𝓍) d𝓍 = 0 ? 

Guida verificata passo dopo passo
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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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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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Percorso guidato
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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 ƒ(x) = x² - 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
Pratica correlata
Domanda del libro di testo

Working with area functions Consider the function ƒ and its graph.

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


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Domanda del libro di testo

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

(c) 1² + 2² + 3² + 4²

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

                                                                                                                                                                    

(c) ∫ sin 2𝓍 d𝓍 = 2 ∫ sin 𝓍 d𝓍 .

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Domanda del libro di testo

{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.


∫₀² (𝓍²―2) d𝓍 ; n = 4

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Domanda del libro di testo

Approximating areas Estimate the area of the region bounded by the graph of ƒ(𝓍) = x² + 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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Domanda del libro di testo

{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.


∫₀^π/2 cos 𝓍 d𝓍 ; n = 4

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