Math Problem Statement
Consider the region bounded by y=x^3, y=0, and x=1. Find the volume of the solid obtained by rotating the region around the vertical line x=2
Solution
We are tasked with finding the volume of the solid formed by rotating the region bounded by , , and around the vertical line . To solve this, we will use the method of cylindrical shells.
Step-by-Step Solution:
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General Setup for Cylindrical Shells: When rotating a region around a vertical line not passing through the region itself, the volume can be found using the formula for the volume of cylindrical shells:
Here:
- The radius of a typical shell is the distance from to the line , which is .
- The height of a typical shell is given by the function .
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Define the Limits of Integration: The region is bounded by and , so the limits of integration are from to .
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Set Up the Integral: Using the formula for the volume of cylindrical shells, we have:
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Simplify the Integrand: Expand the expression inside the integral:
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Integrate: Now integrate term by term:
Simplify the expressions inside the brackets:
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Compute the Final Value: Simplify the expression:
Thus, the volume of the solid is:
Would you like further clarification or a more detailed breakdown of any of the steps?
Related Questions:
- How would the solution change if the rotation was around the line ?
- Can we use the disk/washer method instead of cylindrical shells for this problem?
- What if the region were rotated around the horizontal line ?
- How do we approach this problem if the region is instead bounded by ?
- How do we apply the cylindrical shell method when rotating around ?
Tip: When using the method of cylindrical shells, the radius is always the distance from the rotation axis to the variable , which must be correctly defined depending on the position of the axis.
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Math Problem Analysis
Mathematical Concepts
Calculus
Solid of Revolution
Cylindrical Shell Method
Formulas
V = 2π ∫[a, b] (radius) * (height) dx
Radius = 2 - x
Height = y = x^3
Theorems
Cylindrical Shell Method
Suitable Grade Level
Grades 11-12
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