Textbook Question
9-34. Shell method Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about indicated axis.
{Use of Tech} y = In x/x²,y = 0,x = 3, about the y-axis
9. Graphical Applications of Integrals
Introduction to Volume & Disk Method
5:23 minutesProblem 6.4.38
Textbook Question
35–38. Shell and washer methods Let R be the region bounded by the following curves. Use both the shell method and the washer method to find the volume of the solid generated when R is revolved about the indicated axis.
y = 8,y = 2x+2,x = 0, and x=2; about the y-axis
Verified step by step guidance1
First, identify the region R bounded by the curves: $y = 8$, $y = 2x + 2$, $x = 0$, and $x = 2$. Sketching the region helps visualize the area to be revolved around the y-axis.
For the shell method, consider vertical slices (shells) parallel to the y-axis. The radius of a shell is the distance from the y-axis, which is $x$, and the height of the shell is the difference between the top and bottom functions in terms of $y$, which is $8 - (2x + 2)$.
Set up the integral for the shell method: the volume $V$ is given by $V = \int_{a}^{b} 2\pi \times (\text{radius}) \times (\text{height}) \, dx$. Here, $a=0$ and $b=2$, so write the integral as $V = \int_{0}^{2} 2\pi x (8 - (2x + 2)) \, dx$.
For the washer method, since the solid is revolved about the y-axis, express $x$ in terms of $y$ from the line $y = 2x + 2$, which gives $x = \frac{y - 2}{2}$. The outer radius is the distance from the y-axis to $x=2$, and the inner radius is the distance from the y-axis to $x = \frac{y - 2}{2}$.
Set up the integral for the washer method: the volume $V$ is $V = \int_{c}^{d} \pi \left( R_{outer}^2 - R_{inner}^2 \right) dy$, where $c=8$ and $d=6$ (the y-values corresponding to $x=0$ and $x=2$ on $y=2x+2$). Write the integral as $V = \int_{6}^{8} \pi \left( 2^2 - \left( \frac{y - 2}{2} \right)^2 \right) dy$.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Shell Method
The shell method calculates the volume of a solid of revolution by summing cylindrical shells formed by revolving vertical or horizontal slices around an axis. Each shell's volume is approximated by its circumference times height times thickness. This method is especially useful when the axis of rotation is parallel to the slices.
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Washer Method
The washer method finds volume by integrating cross-sectional areas shaped like washers (disks with holes) perpendicular to the axis of rotation. The volume is the integral of the difference between the outer and inner radii squared, times π. It works well when the solid has a hollow center and the axis of rotation is perpendicular to the slices.
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Setting up the Region and Limits of Integration
Understanding the region bounded by the curves y=8, y=2x+2, x=0, and x=2 is crucial. Identifying the limits of integration and expressing variables in terms of the axis of rotation allows correct setup of integrals for both methods. This step ensures accurate calculation of volume by correctly describing the shape and boundaries.
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