Math Problem Statement
find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis
be aware: rotation about the "x-axis"
shell method with respect to dx
answer should be 512pi/5
full process, thanks
Solution
To find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis using the shell method, follow these steps carefully.
Step 1: Understanding the Given Equation
We are given:
and the boundary condition:
This means we need to solve for values of where:
Step 2: Solving for
Factor the equation:
This gives:
Thus, the region is bounded by and .
Step 3: Shell Method Formula
The shell method formula for rotation about the x-axis is:
- Radius = (distance from the shell to the x-axis)
- Height = (horizontal width of the shell)
- Limits of Integration: to
Thus, the volume integral is:
Step 4: Computing the Integral
Expanding the terms inside the integral:
Now, integrate term by term:
Step 5: Evaluating the Expression
Find a common denominator:
Final Answer:
Do you have any questions, or would you like more details?
Related Questions:
- How does the shell method differ from the disk/washer method?
- What happens if the region is rotated around the y-axis instead?
- How do you determine when to use shells versus washers?
- What if the function were ? How would the volume change?
- How would this problem change if rotated around ?
Tip:
For the shell method, always set up the radius and height carefully, as they depend on the axis of rotation.
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Math Problem Analysis
Mathematical Concepts
Volume of solids of revolution
Shell method
Definite integrals
Formulas
V = 2π ∫ (radius)(height) dy
Integration of polynomial functions: ∫ x^n dx = x^(n+1) / (n+1)
Theorems
Fundamental Theorem of Calculus
Suitable Grade Level
Grades 11-12, Calculus
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