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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 8.8.86
Chapter 8, Problem 8.8.86

Exercises 83–86 are about the infinite region in the first quadrant between the curve y = e^(-x) and the x-axis.
86. Find the volume of the solid generated by revolving the region about the x-axis.

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Identify the region bounded by the curve \(y = e^{-x}\) and the x-axis in the first quadrant. This means \(x \geq 0\) and \(y \geq 0\).
Since the region is revolved about the x-axis, use the disk method to find the volume. The volume of a solid of revolution generated by revolving around the x-axis is given by the integral \(V = \pi \int_a^b [f(x)]^2 \, dx\).
Set up the integral limits. Because the region is in the first quadrant and extends infinitely along the x-axis, the limits are from \(0\) to \(\infty\).
Write the integral for the volume as \(V = \pi \int_0^{\infty} (e^{-x})^2 \, dx = \pi \int_0^{\infty} e^{-2x} \, dx\).
Evaluate the improper integral by finding the antiderivative of \(e^{-2x}\), then apply the limits from \(0\) to \(\infty\) 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 volume can be computed using methods like the disk or washer method, which integrate cross-sectional areas perpendicular to the axis of rotation.
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Finding Volume Using Disks

Disk Method

The disk method calculates volume by slicing the solid into thin disks perpendicular to the axis of rotation. Each disk's volume is approximated by π(radius)^2 times thickness, and integrating these volumes over the interval gives the total volume.
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Disk Method Using y-Axis

Exponential Decay Function y = e^(-x)

The function y = e^(-x) represents exponential decay, approaching zero as x increases. Understanding its behavior is essential for setting integration limits and evaluating the integral that defines the volume of the solid formed by revolving the region bounded by this curve and the x-axis.
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Integrals of Natural Exponential Functions (e^x)
Related Practice
Textbook Question

In Exercises 35–68, use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.

∫ from e to e^e of (ln(ln x) dx)

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Textbook Question

In Exercises 67–73, use integration by parts to establish the reduction formula.

∫ (ln x)^n dx = x (ln x)^n - n ∫ (ln x)^(n-1) dx

Textbook Question

The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.

∫₋₈¹ dx / x^(1/3)

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Textbook Question

In Exercises 35–68, use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.

∫ from 2 to ∞ of (dx / √(x² - 1))

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Textbook Question

The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.

∫₋∞^∞ 2x e^(−x²) dx

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Textbook Question

Evaluate the integrals in Exercises 33–52.

∫ tan⁴(x) sec³(x) dx