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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.15
Chapter 8, Problem 8.AAE.15

Finding volume
The region in the first quadrant enclosed by the coordinate axes, the curve y = e^x, and the line x = 1 is revolved about the y-axis to generate a solid. Find the volume of the solid.

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1
Identify the region bounded by the coordinate axes (x=0 and y=0), the curve \(y = e^{x}\), and the vertical line \(x = 1\) in the first quadrant.
Since the solid is generated by revolving this region about the y-axis, consider using the method of cylindrical shells. The formula for the volume using shells is \(V = \int_{a}^{b} 2\pi \cdot (\text{radius}) \cdot (\text{height}) \, dx\).
In this problem, the radius of a shell at position \(x\) is the distance from the y-axis, which is \(x\), and the height of the shell is the value of the function \(y = e^{x}\).
Set up the integral for the volume as \(V = \int_{0}^{1} 2\pi x e^{x} \, dx\).
To find the volume, evaluate the integral \(\int_{0}^{1} x e^{x} \, dx\) using integration by parts, then multiply the result by \(2\pi\).

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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. Common methods include the disk/washer method and the shell method, which use integration to sum infinitesimal volumes. Choosing the appropriate method depends on the axis of rotation and the shape of the region.
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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. Each shell's volume is approximated by 2π(radius)(height)(thickness). This method is especially useful when revolving around the y-axis and the function is given in terms of x.
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Euler's Method

Exponential Functions and Their Properties

Understanding the function y = e^x is crucial, as it defines the boundary of the region. The exponential function grows rapidly and is continuous and differentiable everywhere. Its inverse and integral properties are often used in setting up and evaluating integrals for volume calculations.
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Properties of Functions
Related Practice
Textbook Question

18. Finding volume (Continuation of Exercise 17.) Find the volume of the solid generated by revolving the region R about:

a. the y-axis.

Textbook Question

Centroid of a region

Find the centroid of the region in the plane enclosed by the curves y = ±(1 − x²)^(-1/2) and the lines x = 0 and x = 1.

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Textbook Question

Evaluate the integrals in Exercises 1–6.

∫ dt / (t - √(1 - t²))

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Textbook Question

Evaluate the integrals in Exercises 1–6.

∫ x arcsin x dx

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.

∫ cos t dt / (1 - cos t)

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.

∫(from π/2 to 2π/3) cos θ dθ / (sin θ cos θ + sin θ)

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