Ch. 8 - Techniques of Integration

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 8 - Techniques of Integration

Problem 8.2.59

Chapter 8, Problem 8.2.59

Finding volume: Find the volume of the solid generated by revolving the region in the first quadrant bounded by the coordinate axes, the curve y = e^x, and the line x = ln(2) about the line x = ln(2).

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Identify the region to be revolved: it is bounded by the coordinate axes (x=0 and y=0), the curve \(y = e^{x}\), and the vertical line \(x = \ln(2)\), all in the first quadrant.

Since the solid is generated by revolving the region about the vertical line \(x = \ln(2)\), consider using the method of cylindrical shells, which is well-suited for rotation around vertical lines.

Set up the volume integral using the shell method. For a typical shell at position \(x\) (where \(0 \leq x \leq \ln(2)\)), the radius of the shell is the horizontal distance from \(x\) to the axis of rotation: \(r = \ln(2) - x\).

The height of the shell is given by the function value at \(x\), which is \(h = e^{x}\). The thickness of the shell is \(dx\).

Write the volume integral as \(V = \int_{0}^{\ln(2)} 2\pi \times (\text{radius}) \times (\text{height}) \, dx = \int_{0}^{\ln(2)} 2\pi (\ln(2) - x) e^{x} \, dx\). This integral can then be evaluated 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 a given axis. Common methods include the disk/washer and shell methods, which use integration to sum infinitesimal volumes. Understanding the shape and axis of rotation is crucial to choosing the appropriate method.

Shell Method

The shell method calculates volume by integrating cylindrical shells formed by revolving vertical or horizontal slices around an axis. Each shell's volume is approximated by circumference × height × thickness. This method is especially useful when the axis of rotation is parallel to the slices and simplifies integration.

Natural Logarithm and Exponential Functions

The problem involves the curve y = e^x and the line x = ln(2). Understanding the properties of exponential and logarithmic functions, including their inverses and domains, helps in setting integration limits and expressing the region accurately for volume calculation.

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Derivative of the Natural Logarithmic Function

Related Practice

Textbook Question

Use the formula ∫ f⁻¹(x) dx = x f⁻¹(x) - ∫ f(y) dy, y = f⁻¹(x)

To evaluate the integrals in Exercises 77-80. Express your answers in terms of x.

∫ arctan x dx

Textbook Question

Solve the initial value problems in Exercises 67–70 for x as a function of t.

(3t⁴ + 4t² + 1) (dx/dt) = 2√3, x(1) = -π√3/4

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Textbook Question

Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.

∫ x arctan(x) dx

Textbook Question

Expand the quotients in Exercises 1–8 by partial fractions.

(2x + 2) / (x² - 2x + 1)

Textbook Question

The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.

∫₁² (8 dx / (x² - 2x + 2))

Textbook Question

Volume: Find the volume of the solid generated by revolving the region in Exercise 45 about the x-axis.

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