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.
∫ (x² / (x² + 1)) dx
Ch. 8 - Techniques of Integration
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 8, Problem 8.8.85
Exercises 83–86 are about the infinite region in the first quadrant between the curve y = e^(-x) and the x-axis.
85. Find the volume of the solid generated by revolving the region about the y-axis.
Verified step by step guidance1
Identify the region bounded by the curve \(y = e^{-x}\), the x-axis, and the y-axis in the first quadrant. This region extends from \(x=0\) to \(x=\infty\) and from \(y=0\) to \(y=1\) since \(e^{-0} = 1\) and \(e^{-\infty} = 0\).
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 cylindrical shells is:
\(V = \int_{a}^{b} 2\pi \cdot (\text{radius}) \cdot (\text{height}) \, dx\)
In this problem, the radius of a shell is the distance from the y-axis, which is \(x\), and the height of the shell is the function value \(y = e^{-x}\). The limits of integration are from \(x=0\) to \(x=\infty\).
Set up the integral for the volume:
\(V = \int_{0}^{\infty} 2\pi x e^{-x} \, dx\)
To evaluate the integral, use integration by parts where you let one part be \(x\) and the other be \(e^{-x}\). After setting up the integral, proceed with integration by parts 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. 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 of the region around a vertical axis. Each shell's volume is approximated by its circumference times height times thickness. This method is especially useful when revolving around the y-axis and the function is given in terms of x.
Recommended video:
07:33Euler's Method
Exponential Decay Function y = e^(-x)
The function y = e^(-x) represents exponential decay, approaching zero as x approaches infinity. Understanding its behavior helps determine the bounds of integration and the shape of the region. Since the region is bounded by this curve and the x-axis in the first quadrant, the limits are from x = 0 to infinity.
Recommended video:
03:39Integrals of Natural Exponential Functions (e^x)
Related Practice
Textbook Question
Evaluate the integrals in Exercises 1–24 using integration by parts.
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Textbook Question
In Exercises 39–48, use an appropriate substitution and then a trigonometric substitution to evaluate the integrals.
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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.
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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.
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Textbook Question
In Exercises 21–32, express the integrand as a sum of partial fractions and evaluate the integrals.
∫ (8x² + 8x + 2) / (4x² + 1)² dx
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