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Ch. 8 - Integration Techniques
Briggs - Calculus: Early Transcendentals 3rd Edition
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All textbooksBriggs 3rd EditionCh. 8 - Integration TechniquesProblem 8.9.91c
Chapter 8, Problem 8.9.91c

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, ∞).
c. Find the minimum value b* such that when b > b*, there exists some a > 0 where A(a,b) = 2.

Verified step by step guidance
1
First, understand the region R is bounded by the curve \(y = e^{-a \cdot x}\) and the x-axis over the interval \([b, \infty)\). The area \(A(a,b)\) of this region is given by the integral of the function from \(b\) to infinity: \[A(a,b) = \int_{b}^{\infty} e^{-a \cdot x} \, dx\]
Next, compute the integral \(A(a,b)\). Since \(a > 0\), the integral converges and can be evaluated as an improper integral: \[A(a,b) = \lim_{t \to \infty} \int_{b}^{t} e^{-a \cdot x} \, dx\] Use the antiderivative of \(e^{-a x}\), which is \(-\frac{1}{a} e^{-a x}\).
Evaluate the definite integral: \[A(a,b) = \lim_{t \to \infty} \left[-\frac{1}{a} e^{-a x} \right]_{x=b}^{x=t} = \lim_{t \to \infty} \left(-\frac{1}{a} e^{-a t} + \frac{1}{a} e^{-a b} \right)\] Since \(a > 0\), \(e^{-a t} \to 0\) as \(t \to \infty\), so the area simplifies to \[A(a,b) = \frac{1}{a} e^{-a b}\]
The problem asks to find the minimum value $b^*$ such that for any $b > b^*$, there exists some \(a > 0\) with \(A(a,b) = 2\). Set up the equation: \[\frac{1}{a} e^{-a b} = 2\] Rewrite it as \[e^{-a b} = 2a\]
To find $b^*$, consider the function \(f(a) = 2a e^{a b}\). For fixed \(b\), the equation \(e^{-a b} = 2a\) can be rearranged to \(1 = 2a e^{a b}\). We want to find the smallest \(b\) such that this equation has a positive solution \(a\). Analyze the function \(g(a) = 2a e^{a b}\) for \(a > 0\) and find its minimum value with respect to \(a\). Then, find $b^*$ such that the minimum of \(g(a)\) equals 1. This involves taking the derivative of \(g(a)\) with respect to \(a\), setting it to zero to find critical points, and solving for \(b\) in terms of \(a\). This will give the minimal $b^*$.

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Key Concepts

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

Definite Integral and Area Under a Curve

The definite integral calculates the area between a curve and the x-axis over a specified interval. For y = e^{-a·x}, integrating from b to infinity gives the total area under the exponential decay curve, which is essential for expressing A(a,b) and analyzing its behavior.
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Definition of the Definite Integral

Improper Integrals and Convergence

Integrals with infinite limits, like from b to ∞, are improper and require evaluating limits to determine convergence. Understanding when the integral converges and how it depends on parameters a and b is crucial to finding values where the area equals a specific number.
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Improper Integrals: Infinite Intervals

Parameter Dependence and Optimization

The problem involves finding a minimum b* such that for b > b*, there exists an a > 0 making the area equal to 2. This requires analyzing how the integral's value changes with parameters a and b, and using algebraic or calculus methods to solve for the critical threshold b*.
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Intro to Applied Optimization: Maximizing Area
Related Practice
Textbook Question

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Textbook Question

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c. A polynomial that fits the data reasonably well is:

g(t) = 3.49t³ - 43.21t² + 142.43t - 1.75

Estimate the elevation of the balloon after five minutes using this polynomial.

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Textbook Question

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Textbook Question

Prove the following orthogonality relations (which are used to generate Fourier series). Assume m and n are integers with m ≠ n.

c.

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Textbook Question

3. What term(s) should appear in the partial fraction decomposition of a proper rational function with each of the following?

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