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Chapter 1: Problem 64

Berechnen Sie das Volumen der K?rper, die durch Rotation der folgenden Meridiane entstehen: a) \(y=\frac{1}{\cosh x},-2 \leqq x \leq 2\), Rotation um die \(x\)-Achse b) \((y-2)^{2}+x^{2}=1\) Rotation um dic \(x\)-Achse c) \((y-3)^{2}+3 x=0,-3 \leq x \leqq 0\) Rotation um die \(x\)-Achse d) \(y=\sin x, 0 \leqq x \leqq 2 \pi\) Rotation um dic \(x\)-Achse e) \(x^{2}-y^{2}-4,-2 \leqq y \leqq 2\) Rotation um die \(y\)-Achse f) \(y=\frac{1}{\cos x}\) mit \(-\frac{\pi}{4}

Short Answer

Expert verified
The volume for part (a) is approximately 6.1543 cubic units.

Step by step solution

01

Understand Volume of Rotation Concepts

When a curve is rotated around the x-axis or y-axis, it forms a 3-D solid. The volume of this solid can be calculated using the disk or washer method, depending on the problem statement and the axis of rotation.
02

Part a - Establish the Function and Limits

For part a, we have the function \(y = \frac{1}{\cosh x}\) and the limits \(-2 \leq x \leq 2\). We are rotating about the x-axis.
03

Part a - Use Disk Method Formula

The volume of a solid obtained by rotating the curve \(y = f(x)\) around the x-axis from \(x = a\) to \(x = b\) is given by \[ V = \pi \int_{a}^{b} [f(x)]^2 \, dx \] Apply this to our function: \[ V = \pi \int_{-2}^{2} \left(\frac{1}{\cosh x}\right)^2 \, dx \].
04

Part a - Integrate Using Hyperbolic Identity

Use the identity \( \sech x = \frac{1}{\cosh x}\) and notice that \( \sech^2 x = 1 - \tanh^2 x\). The integration will use trigonometric substitution or numeric methods for precise calculation:\[ \int \sech^2 x \, dx = \tanh x \]Thus, the integral becomes \[ V = \pi \left[ \tanh x \right]_{-2}^{2} \].
05

Part a - Evaluate the Integral

Evaluate \( \tanh 2 - \tanh(-2) \). Numerical evaluation gives:\[ \tanh 2 \approx 0.964 \] \[ \tanh(-2) = -\tanh 2 \approx -0.964 \] Thus, \[ V \approx \pi (0.964 - (-0.964)) \approx \pi \times 1.928 \].
06

Part a - Calculate Final Volume

The final volume after computation is \( V \approx 6.1543 \) cubic units.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Disk Method
The Disk Method is a fundamental integration technique to determine the volume of a solid of revolution. It is primarily used when a region in the plane is rotated about an axis, typically the x-axis or y-axis, forming a solid. To visualize this, imagine stacking a series of thin, circular disks along the axis. The thickness of each disk is an infinitesimally small change in x (or y), and the radius is determined by the value of the function at a particular point.
  • When rotating a function around the x-axis, use the formula: \[ V = \pi \int_{a}^{b} [f(x)]^2 \, dx \]
  • For a rotation around the y-axis, adjust the formula to correspond to the y variable.
The Disk Method is straightforward when the region being rotated is contiguous and aligned in a way that forms a perfect disk for each slice.
Washer Method
The Washer Method is an extension of the Disk Method and is used when the solid formed has a hole in it, much like a "washer." Instead of stacking solid disks, you stack washers with empty centers. It's applied when there is another inner function that creates a cavity within the rotated solid, forming a hollow space.
  • To apply, subtract the volume of the inner function's disk from the outer one:\[ V = \pi \int_{a}^{b} ([f(x)]^2 - [g(x)]^2) \, dx \]
  • The function defining the outer radius is \(f(x)\), and the inner function \(g(x)\) represents the cavity's shape.
The Washer Method is particularly useful for problems where the region is bounded by two curves which, when rotated, form a washer-like shape.
Hyperbolic Functions
Hyperbolic functions are analogs of trigonometric functions but for exponential functions. The functions most commonly encountered are hyperbolic sine (\( \sinh x \)) and hyperbolic cosine (\( \cosh x \)). They are defined as:
  • \( \sinh x = \frac{e^x - e^{-x}}{2} \)
  • \( \cosh x = \frac{e^x + e^{-x}}{2} \)
These functions are particularly useful in calculus and related areas of mathematics, offering unique properties for solving certain integral and differential equations.
For calculating volumes of revolutions involving \( \cosh x \), understanding its reciprocal, \( \sech x \), plays a crucial role, particularly when paired with its identity \( \sech^2 x = 1 - \tanh^2 x \).
Integration Techniques
Integration techniques are essential tools for solving the integrals involved in determining volumes of rotation. They include substitution, integration by parts, and partial fraction decomposition, among others. In particular, for the volume of revolution problems:
  • Substitution: Useful for simplifying integrals by changing the variable of integration, especially helpful with trigonometric or hyperbolic identities.
  • Numerical Methods: Applied when the integral does not resolve easily using standard techniques. Methods like the trapezoidal rule or Simpson's rule might be deployed for approximation.
These techniques help to evaluate the integral parts of the disk or washer method formulas, transforming complex integrals into more manageable computations.
Rotation around X-axis
Rotating a curve around the x-axis forms a three-dimensional solid, a common application in calculus. The x-axis serves as the line of rotation, with the endpoint values providing the bounds of the rotation. Constructing the solid involves:
  • Visualizing the curve as it sweeps around the x-axis, creating a symmetrical shape.
  • Applying either the Disk or Washer Method to determine the volume.
The steps for rotation involve setting up the proper integral and ensuring the function's radius accurately reflects its distance from the axis of rotation. This rotation usually simplifies calculation as the resulting shape often has predictable properties, like circular or rotational symmetry.

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