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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 8.9.91a
Chapter 8, Problem 8.9.91a

91. [Use of Tech] Regions bounded by exponentials Let a > 0 and let R be the region bounded by the graph of y = e^(-a·x) and the x-axis
on the interval [b, ∞).
a. Find A(a,b), the area of R as a function of a and b.

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1
Identify the region R bounded by the curve \(y = e^{-a \cdot x}\) and the x-axis on the interval \([b, \infty)\). Since the curve is above the x-axis for \(a > 0\), the area can be found by integrating the function from \(x = b\) to \(x = \infty\).
Set up the integral for the area \(A(a,b)\) as: \(A(a,b) = \int_{b}^{\infty} e^{-a \cdot x} \, dx\)
Recall the integral formula for the exponential function: \(\int e^{kx} \, dx = \frac{1}{k} e^{kx} + C\), where \(k\) is a constant. Here, \(k = -a\).
Evaluate the definite integral by applying the limits from \(b\) to \(\infty\): \(A(a,b) = \left[ \frac{e^{-a \cdot x}}{-a} \right]_{b}^{\infty}\)
Calculate the limit as \(x \to \infty\) of \(e^{-a \cdot x}\), which approaches 0 for \(a > 0\), and then substitute \(x = b\) to express the area \(A(a,b)\) in terms of \(a\) and \(b\).

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Key Concepts

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

Definite Integrals and Area Under a Curve

The area under a curve between two points is found using definite integrals. For a function f(x), the area from x = b to x = c is the integral of f(x) dx over [b, c]. When the upper limit is infinity, improper integrals are used to evaluate the area.
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Definition of the Definite Integral

Exponential Functions and Their Properties

Exponential functions like y = e^(-a·x) decay rapidly as x increases when a > 0. Understanding their behavior helps in setting up integrals and evaluating limits, especially for infinite intervals where the function approaches zero.
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Properties of Functions

Improper Integrals and Convergence

When integrating over an infinite interval, the integral is called improper. To find the area, one must evaluate the limit of the integral as the upper bound approaches infinity and verify that this limit converges to a finite value.
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Improper Integrals: Infinite Intervals
Related Practice
Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

a. ∫(3/(x² + 4)) dx = ∫(3/x²) dx + ∫(3/4) dx.

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Textbook Question

Arc length of a parabola Let L(c) be the length of the parabola f(x) = x² from x = 0 to x = c, where c ≥ 0 is a constant.

a. Find an expression for L.

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Textbook Question

60. Two Methods

a. Evaluate ∫(x · ln(x²)) dx using the substitution u = x² and evaluating ∫(ln(u)) du.

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Textbook Question

Gamma function The gamma function is defined by Γ(p) = ∫ from 0 to ∞ of x^(p-1) e^(-x) dx, for p not equal to zero or a negative integer.

a. Use the reduction formula ∫ from 0 to ∞ of x^p e^(-x) dx = p ∫ from 0 to ∞ of x^(p-1) e^(-x) dx for p = 1, 2, 3, ...

to show that Γ(p + 1) = p! (p factorial).

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Textbook Question

65. Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

a. To evaluate ∫ (4x⁶)/(x⁴ + 3x²) dx, the first step is to find the partial fraction decomposition of the integrand.

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Textbook Question

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

70. Let f(x) = e^(-x²).

a. Find a Simpson's Rule approximation to the integral from 0 to 3 of e^(-x²) dx using n = 30 subintervals.

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