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Ch. 8 - Integration Techniques
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 8 - Integration TechniquesProblem 8.4.78b
Chapter 8, Problem 8.4.78b

Computing areas On the interval [0,2], the graphs of f(x)=x²/3 and g(x)=x²(9−x²)^(-1/2) have similar shapes.
b. Find the area of the region bounded by the graph of g and the x-axis on the interval [0,2].

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1
Identify the function and the interval for which you need to find the area. Here, the function is \(g(x) = \frac{x^{2}}{\sqrt{9 - x^{2}}}\) and the interval is \([0, 2]\).
Recall that the area under the curve of a function \(g(x)\) from \(a\) to \(b\) is given by the definite integral \(\int_{a}^{b} g(x) \, dx\). So, set up the integral for the area as \(\int_{0}^{2} \frac{x^{2}}{\sqrt{9 - x^{2}}} \, dx\).
Consider a substitution to simplify the integral. Since the denominator involves \(\sqrt{9 - x^{2}}\), a trigonometric substitution such as \(x = 3 \sin \theta\) is appropriate because it will simplify the square root expression.
Perform the substitution: express \(x^{2}\), \(dx\), and \(\sqrt{9 - x^{2}}\) in terms of \(\theta\). For example, \(x = 3 \sin \theta\) implies \(dx = 3 \cos \theta \, d\theta\) and \(\sqrt{9 - x^{2}} = 3 \cos \theta\). Also, update the limits of integration accordingly by substituting \(x=0\) and \(x=2\) into \(x=3 \sin \theta\) to find the new \(\theta\) limits.
Rewrite the integral in terms of \(\theta\), simplify the integrand, and then integrate with respect to \(\theta\). After integrating, substitute back to \(x\) if necessary to express the area in terms of the original variable.

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Key Concepts

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

Definite Integral for Area Calculation

The definite integral of a function over an interval [a, b] represents the net area between the graph of the function and the x-axis. When the function is non-negative on the interval, the integral gives the exact area under the curve. This concept is essential for finding the area bounded by g(x) and the x-axis on [0, 2].
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Definition of the Definite Integral

Integrating Functions with Radical Expressions

Functions involving radicals, such as g(x) = x² / √(9 - x²), often require substitution methods for integration. Recognizing the form and applying appropriate substitutions, like trigonometric substitution, simplifies the integral and makes it solvable. This technique is crucial for handling the integral of g(x).
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Limits of Rational Functions with Radicals

Domain and Interval Considerations

Understanding the domain of the function and the interval of integration ensures the integral is properly set up. Since g(x) involves a square root in the denominator, the expression under the root must be positive, restricting the domain. Confirming the function is defined and continuous on [0, 2] is necessary before integrating.
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Related Practice
Textbook Question

2. Give an example of each of the following.

b. A repeated linear factor

Textbook Question

58. Two Methods Evaluate ∫(from 0 to π/3) sin(x) · ln(cos(x)) dx in the following two ways:

b. Use substitution.

Textbook Question

The Eiffel Tower Property Let R be the region between the curves y = e^(-c·x) and y = -e^(-c·x) on the interval [a, ∞), where a ≥ 0 and c > 0.

The center of mass of R is located at (x̄, 0), where x̄ = [∫(a to ∞) x·e^(-c·x) dx] / [∫(a to ∞) e^(-c·x) dx]

(The profile of the Eiffel Tower is modeled by these two exponential curves; see the Guided Project ""The exponential Eiffel Tower"")

b. With a = 0 and c = 2, find the equations of the lines tangent to both curves at x = 0

Textbook Question

45–48. {Use of Tech} Trapezoid Rule and Simpson’s Rule Consider the following integrals and the given values of n.

45. ∫(0 to 1) e^(2x) dx; n = 25

c. Compute the absolute errors in the Trapezoid Rule and Simpson’s Rule with 2n subintervals.

Textbook Question

59. Two Methods

b. Evaluate ∫(x / √(x + 1)) dx using substitution.

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Textbook Question

66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in the following exercises.

71. Let f(x) = √(sin x).

b. Find an upper bound on the absolute error in the estimate from part (a) using Theorem 8.1. (Hint: |f''''(x)| ≤ 1 on [1,2].)