Math Problem Statement
Solution
The problem involves calculating the volume of the solid of revolution obtained by rotating the given region about the -axis. The region is defined by the curve:
and bounded by and the vertical line .
Steps to solve:
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Volume formula for revolution about the -axis: The formula for the volume of a solid of revolution about the -axis is: where is the function describing the curve, and and are the bounds of .
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Set up the integral:
- The bounds are from to .
- The function is .
- The square of the function is:
Substituting this into the volume formula:
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Expand the square: Expand :
So the integral becomes:
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Integrate term by term: Compute each term separately:
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Add the results: Combine all terms:
First, simplify the fractions:
Combine:
-
Simplify:
Reduce the fraction:
Final Answer:
Would you like detailed steps or have any questions?
Related Questions:
- How do you handle the volume of revolution when the region is bounded by two curves?
- How does the disk method differ from the shell method?
- What if the region is revolved around a vertical axis instead?
- Can this problem be solved using numerical integration instead?
- How does symmetry simplify volume of revolution problems?
Tip:
Always double-check your expanded polynomial terms to avoid calculation errors!
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Math Problem Analysis
Mathematical Concepts
Volumes of Revolution
Definite Integrals
Polynomial Expansion
Formulas
Volume of revolution about x-axis: V = π ∫[a to b] [f(x)]^2 dx
Polynomial expansion: (a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca
Theorems
Disk Method for Volumes of Revolution
Suitable Grade Level
Grade 11-12 or introductory college-level calculus
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