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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 8.2.60a
Chapter 8, Problem 8.2.60a

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 = 1.
a. About the y-axis.

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1
Identify the region to be revolved: it is bounded by the x-axis (y=0), y-axis (x=0), the curve \(y = e^{-x}\), and the vertical line \(x = 1\) in the first quadrant.
Since the solid is generated by revolving the region about the y-axis, consider using the method of cylindrical shells. The formula for the volume using shells is \(V = \int_a^b 2\pi \cdot (\text{radius}) \cdot (\text{height}) \, dx\).
Determine the radius and height of a typical shell: the radius is the distance from the y-axis, which is \(x\), and the height is the value of the function \(y = e^{-x}\).
Set up the integral for the volume: \(V = \int_0^1 2\pi x e^{-x} \, dx\).
Evaluate the integral using integration by parts or an appropriate method to find the volume (do not compute the final value here).

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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 integral methods such as the disk/washer or shell method, depending on the axis of rotation and the shape of the region.
Recommended video:
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Finding Volume Using Disks

Shell Method

The shell method calculates volume by integrating cylindrical shells formed by revolving vertical slices around a vertical axis. It is especially useful when rotating around the y-axis and when the function is given in terms of x, simplifying the integral setup.
Recommended video:
07:33
Euler's Method

Exponential Functions and Their Properties

Understanding the behavior of the function y = e^(-x) is crucial, as it defines the boundary of the region. Knowing its decay and how to integrate expressions involving e^(-x) helps in setting up and evaluating the integral for volume.
Recommended video:
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Properties of Functions
Related Practice
Textbook Question

In Exercises 11–22, estimate the minimum number of subintervals needed to approximate the integrals with an error of magnitude less than 10^-4 by (a) the Trapezoidal Rule (The integrals in Exercises 11–18 are the integrals from Exercises 1–8.)

∫ from 2 to 4 of 1/(s - 1)² ds

Textbook Question

Lifetime of a tire Assume the random variable L in Example 2f is normally distributed with mean μ = 22,000 miles and σ = 4,000 miles.

a. In a batch of 4000 tires, how many can be expected to last for at least 18,000 miles?

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Textbook Question

In Exercises 11–22, estimate the minimum number of subintervals needed to approximate the integrals with an error of magnitude less than 10^-4 by (a) the Trapezoidal Rule (The integrals in Exercises 11–18 are the integrals from Exercises 1–8.)

∫ from 1 to 3 of (2x - 1) dx

Textbook Question

In Exercises 11–22, estimate the minimum number of subintervals needed to approximate the integrals with an error of magnitude less than 10^-4 by (a) the Trapezoidal Rule (The integrals in Exercises 11–18 are the integrals from Exercises 1–8.)

∫ from 0 to 2 of sin(x + 1) dx

Textbook Question

In Exercises 11–22, estimate the minimum number of subintervals needed to approximate the integrals with an error of magnitude less than 10^-4 by (a) the Trapezoidal Rule (The integrals in Exercises 11–18 are the integrals from Exercises 1–8.)

∫ from 0 to 3 of 1/√(x + 1) dx

Textbook Question

Centroid:

Find the centroid of the region cut from the first quadrant by the curve

y = 1/√(x + 1) and the line x = 3.