Textbook Question
For the following regions R, determine which is greater—the volume of the solid generated when R is revolved about the x-axis or about the y-axis.
R is bounded by y=1−x^3, the x-axis, and the y-axis.
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9. Graphical Applications of Integrals
Introduction to Volume & Disk Method
5:49 minutesProblem 6.3.58
Textbook Question
Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given line.
y=x and y=1+x/2; about y=3
Verified step by step guidance1
First, identify the region R bounded by the curves \( y = x \) and \( y = 1 + \frac{x}{2} \). To do this, find the points of intersection by setting the two equations equal: \( x = 1 + \frac{x}{2} \).
Solve the equation \( x = 1 + \frac{x}{2} \) for \( x \) to find the limits of integration. This will give the \( x \)-values where the two curves intersect, which define the interval over which the region R lies.
Since the solid is generated by revolving the region about the line \( y = 3 \), use the washer method. The radius of each washer is the distance from the line \( y = 3 \) to the curve. Express the outer radius \( R(x) \) and inner radius \( r(x) \) as functions of \( x \):
\[ R(x) = 3 - y_{\text{lower}}(x), \quad r(x) = 3 - y_{\text{upper}}(x) \]
where \( y_{\text{lower}}(x) \) and \( y_{\text{upper}}(x) \) are the lower and upper curves respectively on the interval.
Set up the volume integral using the washer method formula:
\[ V = \pi \int_{a}^{b} \left[ R(x)^2 - r(x)^2 \right] \, dx \]
where \( a \) and \( b \) are the intersection points found in step 2.
Evaluate the integral to find the volume of the solid. This involves squaring the radii expressions, subtracting, and integrating with respect to \( x \) over the interval \( [a, b] \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Finding the Region Bounded by Curves
To find the volume of a solid generated by revolving a region, first identify the area bounded by the given curves. This involves solving for the points of intersection and understanding which curve lies above or below within the interval. Here, the region is bounded by y = x and y = 1 + x/2.
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Volume of Solids of Revolution Using the Washer Method
When a region is revolved around a horizontal line not on the x-axis, the volume can be found using the washer method. This involves integrating the difference of the squares of the outer and inner radii (distances from the axis of rotation to the curves) with respect to x or y, depending on the setup.
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Adjusting Radii for Revolution About a Line y = k
When revolving around a horizontal line y = k, the radius of each washer is the vertical distance between the curve and the line y = k. This requires calculating |k - y| for each curve to determine the inner and outer radii, ensuring correct setup of the integral for volume.
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