Math Problem Statement
Solution
To solve this problem, we need to use the shell method for finding the volume of the solid. Let's solve each part step by step:
Part (a)
The formula for the shell method involves the radius, , and the height, . From the problem:
- The height of the shell is given by the function , which is the boundary of the region.
So, the height .
Part (b)
The volume using the shell method is given by: where , and is the upper limit of integration (where intersects ).
To find , solve : Thus, or . So, .
The volume formula becomes:
Part (c)
Simplify the integrand: So:
Now compute the integral:
Evaluate at the limits: At :
Find a common denominator (20):
So:
Final Answer:
Let me know if you have any questions or want further details!
Questions to explore:
- How does the shell method differ from the disk/washer method?
- Why is used for height in this problem?
- Can the same volume be computed using the disk/washer method? If so, how?
- What role does symmetry play in simplifying the integration?
- How would the integral change if the region was rotated about a different axis?
Tip: Always double-check the limits of integration when solving problems involving revolutions. Misidentifying limits can lead to significant errors.
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Math Problem Analysis
Mathematical Concepts
Volume of solids of revolution
Shell method
Definite integration
Formulas
Shell method formula: V = 2π ∫[a to b] x f(x) dx
Theorems
Fundamental Theorem of Calculus
Suitable Grade Level
Grades 11-12
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