Textbook Question
Determine the area of the shaded region in the following figures.
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Ch. 6 - Applications of Integration
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 6, Problem 6.4.61
53–62. Choose your method Let R be the region bounded by the following curves. Use the method of your choice to find the volume of the solid generated when R is revolved about the given axis.
y = x²,y=2−x, and x=0, in the first quadrant; about the y-axis
Verified step by step guidance1
First, identify the region R bounded by the curves: \(y = x^{2}\), \(y = 2 - x\), and \(x = 0\) in the first quadrant. Sketching these curves helps visualize the area and the solid formed when revolving around the y-axis.
Determine the points of intersection between \(y = x^{2}\) and \(y = 2 - x\) by setting \(x^{2} = 2 - x\). Solve this equation to find the x-values where the curves meet, which will serve as limits of integration.
Since the solid is generated by revolving the region around the y-axis, decide on the method to use: the shell method is often convenient when revolving around the y-axis and the functions are given in terms of x.
Set up the volume integral using the shell method formula: \(V = \int_{a}^{b} 2\pi \cdot (\text{radius}) \cdot (\text{height}) \, dx\). Here, the radius is the distance from the y-axis, which is \(x\), and the height is the vertical distance between the curves, \( (2 - x) - x^{2} \).
Write the integral explicitly as \(V = \int_{x=a}^{x=b} 2\pi x \left( (2 - x) - x^{2} \right) dx\), where \(a\) and \(b\) are the intersection points found earlier. This integral represents the volume of the solid formed by revolving the region around the y-axis.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Setting up the region bounded by curves
Understanding the region R requires identifying the area enclosed by the curves y = x², y = 2 - x, and the line x = 0 in the first quadrant. This involves finding the points of intersection and sketching the region to visualize the limits of integration.
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Finding Area Between Curves on a Given Interval
Methods of finding volumes of solids of revolution
Volumes generated by revolving a region around an axis can be found using methods like the disk/washer method or the shell method. Choosing the appropriate method depends on the axis of rotation and the shape of the region to simplify integration.
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04:48Finding Volume Using Disks
Volume calculation about the y-axis using the shell method
When revolving around the y-axis, the shell method is often convenient. It involves integrating cylindrical shells with radius equal to the x-value and height given by the difference of the functions, integrating with respect to x over the region's domain.
Recommended video:
06:30Disk Method Using y-Axis
Related Practice
Textbook Question
Orientation and force A plate shaped like an equilateral triangle 1 m on a side is placed on a vertical wall 1 m below the surface of a pool filled with water. On which plate in the figure is the force greater? Try to anticipate the answer and then compute the force on each plate.
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Textbook Question
53–62. Choose your method Let R be the region bounded by the following curves. Use the method of your choice to find the volume of the solid generated when R is revolved about the given axis.
y = x² and y = 2−x²; about the x-axis
52
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Textbook Question
53–62. Choose your method Let R be the region bounded by the following curves. Use the method of your choice to find the volume of the solid generated when R is revolved about the given axis.
y = x−x⁴,y=0; about the x-axis.
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Textbook Question
Find the area of the region described in the following exercises.
The region bounded by y=|x−3|and y=x/2
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Textbook Question
Find the area of the region described in the following exercises.
The region bounded by y=2 / 1 + x^2 and y=1
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