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Ch. 8 - Techniques of Integration
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 8 - Techniques of IntegrationProblem 8.AAE.17a
Chapter 8, Problem 8.AAE.17a

Finding volume
Let R be the "triangular" region in the first quadrant that is bounded above by the line y = 1, below by the curve y = ln x, and on the left by the line x = 1.
Find the volume of the solid generated by revolving R about
a. the x-axis.

Verified step by step guidance
1
First, identify the region R bounded by the curves: above by \(y = 1\), below by \(y = \ln x\), and on the left by \(x = 1\). Since \(y = \ln x\) is defined for \(x > 0\), and the region is in the first quadrant, determine the right boundary by finding the \(x\)-value where \(y = 1\) intersects \(y = \ln x\). Solve \(\ln x = 1\) to find this \(x\)-value.
Set up the volume integral using the method of washers (disks with holes) since the region is revolved around the x-axis. The volume element is given by \(\pi \times (\text{outer radius}^2 - \text{inner radius}^2) \, dx\) where the outer radius is the distance from the x-axis to the upper curve and the inner radius is the distance from the x-axis to the lower curve.
In this problem, the outer radius is the distance from the x-axis to \(y = 1\), which is simply 1, and the inner radius is the distance from the x-axis to \(y = \ln x\). So the volume integral becomes \(V = \int_{x=1}^{x=e} \pi \left(1^2 - (\ln x)^2\right) \, dx\) where \(e\) is the solution from step 1.
Write the integral explicitly as \(V = \pi \int_1^{e} \left(1 - (\ln x)^2\right) \, dx\). This integral represents the volume of the solid generated by revolving the region R about the x-axis.
To solve the integral, split it into two parts: \(\pi \int_1^{e} 1 \, dx - \pi \int_1^{e} (\ln x)^2 \, dx\). The first integral is straightforward, and the second integral can be solved using integration by parts or a known formula for \(\int (\ln x)^n \, dx\). After evaluating both integrals, combine the results to express the volume.

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Key Concepts

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

Region Bounded by Curves

Understanding the region R requires identifying the area enclosed by the given curves and lines: y = 1 (a horizontal line), y = ln(x) (a logarithmic curve), and x = 1 (a vertical line). This helps determine the limits of integration and the shape of the region to be revolved.
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Finding Area When Bounds Are Not Given

Volume of Solids of Revolution

When a region is revolved around an axis, it generates a 3D solid. The volume can be found using methods like the disk/washer method, which involves integrating cross-sectional areas perpendicular to the axis of revolution.
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Finding Volume Using Disks

Disk/Washer Method

This method calculates volume by slicing the solid into thin disks or washers. For revolution about the x-axis, the radius of each disk is the vertical distance from the axis to the curve, and the volume is the integral of π times the radius squared over the interval.
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Disk Method Using y-Axis
Related Practice
Textbook Question

Use the substitutions in Equations (1)–(4) to evaluate the integrals in Exercises 33–40. Integrals like these arise in calculating the average angular velocity of the output shaft of a universal joint when the input and output shafts are not aligned.

∫ dx / (1 + sin x + cos x)

Textbook Question

Finding volume

The infinite region bounded by the coordinate axes and the curve y = −ln x in the first quadrant is revolved about the x-axis to generate a solid. Find the volume of the solid.

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Textbook Question

Finding volume

The region in the first quadrant that is enclosed by the x-axis and the curve y = 3x√(1 − x) is revolved about the y-axis to generate a solid. Find the volume of the solid.

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Textbook Question

Evaluate the integrals in Exercises 1–6.

∫ dt / (t - √(1 - t²))

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Textbook Question

For each x > 0, let G(x) = ∫(from 0 to x) e^(-xt) dt. Prove that xG(x) = 1 for each x > 0.

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Textbook Question

Use the substitution z = tan(θ/2) to evaluate the integrals in Exercises 41 and 42.

∫ csc θ dθ