Textbook Question
In Exercises 59–86, find the derivative of y with respect to the given independent variable.
73. y = log₄ x + log₄ x²
Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 7, Problem 7.7.80
80. Volume The region enclosed by the curve y=sech(x), the x-axis, and the lines x=±ln√3 is revolved about the x-axis to generate a solid. Find the volume of the solid.
Verified step by step guidance1
Identify the region to be revolved: it is bounded by the curve \(y = \text{sech}(x)\), the x-axis (\(y=0\)), and the vertical lines \(x = -\ln\sqrt{3}\) and \(x = \ln\sqrt{3}\).
Recall the formula for the volume of a solid of revolution about the x-axis using the disk method: \(V = \pi \int_{a}^{b} [f(x)]^2 \, dx\), where \(f(x)\) is the function being revolved.
Set up the integral for the volume: \(V = \pi \int_{-\ln\sqrt{3}}^{\ln\sqrt{3}} \left( \text{sech}(x) \right)^2 \, dx\).
Use the identity \(\text{sech}(x) = \frac{1}{\cosh(x)}\), so the integrand becomes \(\text{sech}^2(x) = \frac{1}{\cosh^2(x)}\).
Evaluate the integral \(\int \text{sech}^2(x) \, dx\), which is a standard integral with antiderivative \(\tanh(x)\), and then apply the limits \(x = -\ln\sqrt{3}\) to \(x = \ln\sqrt{3}\) to find the volume.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Volume of Solids of Revolution
This concept involves finding the volume of a 3D solid formed by rotating a 2D region around an axis. The disk or washer method is commonly used, where the volume is computed by integrating the cross-sectional area perpendicular to the axis of rotation.
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Finding Volume Using Disks
Hyperbolic Secant Function (sech(x))
The hyperbolic secant function, sech(x), is defined as 1/cosh(x), where cosh(x) = (e^x + e^{-x})/2. Understanding its properties and behavior is essential for setting up the integral limits and integrand when calculating the volume.
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6:22Graphs of Secant and Cosecant Functions
Definite Integration with Logarithmic Limits
The problem involves integrating between x = ±ln(√3), which requires understanding how to handle logarithmic limits in definite integrals. Proper evaluation of the integral at these bounds is crucial for obtaining the exact volume.
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05:43Definition of the Definite Integral
Related Practice
Textbook Question
Evaluate the integrals in Exercises 87–96.
95. ∫₂⁴ x^(2x) (1 + ln x) dx
1
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Textbook Question
Evaluate the integrals in Exercises 87–96.
87. ∫ 5ˣ dx
Textbook Question
In Exercises 13–24, find the derivative of y with respect to the appropriate variable.
15. y = 2√t tanh(√t)
Textbook Question
Evaluate the integrals in Exercises 41–60.
49. ∫(sech(√t)tanh(√t)dt)/√t
Textbook Question
In Exercises 115–126, use logarithmic differentiation or the method in Example 6 to find the derivative of y with respect to the given independent variable.
120. y = x^(sin x)