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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 5.5.98
Chapter 5, Problem 5.5.98

Areas of regions Find the area of the following regions.                                                                                                                   
                                                                                                                                                                 The region bounded by the graph of Ζ’(𝓍) = (𝓍―4)⁴ and the 𝓍-axis between and 𝓍 = 2 and π“= 6

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Identify the function and the interval: The function is given as \(f(x) = (x - 4)^4\), and the region is bounded between \(x = 2\) and \(x = 6\) along the \(x\)-axis.
Determine the nature of the function on the interval: Since \((x - 4)^4\) is always non-negative for all real \(x\), the graph lies above or on the \(x\)-axis between \(x=2\) and \(x=6\).
Set up the definite integral to find the area: The area under the curve and above the \(x\)-axis from \(x=2\) to \(x=6\) is given by the integral \(\int_{2}^{6} (x - 4)^4 \, dx\).
Apply the power rule for integration: Recall that \(\int (x - a)^n \, dx = \frac{(x - a)^{n+1}}{n+1} + C\). Use this to write the antiderivative of \((x - 4)^4\).
Evaluate the definite integral: Substitute the upper limit \(x=6\) and the lower limit \(x=2\) into the antiderivative expression and find the difference to get the area.

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Key Concepts

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

Definite Integral as Area Under a Curve

The definite integral of a function between two points gives the net area between the graph and the x-axis over that interval. For functions above the x-axis, this corresponds to the actual area. Calculating this integral helps find the exact area of the region bounded by the curve and the axis.
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Definition of the Definite Integral

Properties of Even-Powered Polynomial Functions

Functions like Ζ’(x) = (x - 4)^4 are always non-negative because even powers eliminate negative values. This ensures the graph lies on or above the x-axis, simplifying area calculations since the integral directly represents the area without needing to consider negative parts.
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Properties of Functions

Setting Integration Limits Based on the Region

The limits of integration correspond to the x-values that bound the region of interest. Here, the area is between x = 2 and x = 6, so these values are used as the lower and upper limits in the definite integral to calculate the area under the curve within this interval.
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Integration by Parts for Definite Integrals Example 8
Related Practice
Textbook Question

Max/min of area functions Suppose Ζ’ is continuous on [0 ,∞) and A(𝓍) is the net area of the region bounded by the graph of Ζ’ and the t-axis on [0, x]. Show that the local maxima and minima of A occur at the zeros of Ζ’. Verify this fact with the function Ζ’(𝓍) = 𝓍² - 10𝓍.

Textbook Question

{Use of Tech} Sigma notation for Riemann sums Use sigma notation to write the following Riemann sums. Then evaluate each Riemann sum using Theorem 5.1 or a calculator.

The right Riemann sum for Ζ’(𝓍)) = x + 1 on [0, 4] with n = 50.

Textbook Question

Definite integrals Use geometry (not Riemann sums) to evaluate the following definite integrals. Sketch a graph of the integrand, show the region in question, and interpret your result.                                                                                                                                      

                                                                                                                                                                                       

 βˆ«β‚€β΄ (8―2𝓍) d𝓍

Textbook Question

Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals.                                                                                                                         

                                                                                                                                                                              

 βˆ«Ο€/₁₆^Ο€/⁸ 8 cscΒ² 4𝓍 d𝓍

Textbook Question

Average values Find the average value of the following functions on the given interval. Draw a graph of the function and indicate the average value.

Ζ’(𝓍) = 1/(𝓍² + 1) on [―1, 1]

Textbook Question

Integrals with sinΒ² 𝓍 and cosΒ² 𝓍 Evaluate the following integrals.                                                                                                             

                                                                                                                                                                    

 βˆ«β‚€^Ο€/⁴ cosΒ² 8ΞΈ dΞΈ