Math Problem Statement
Solution
The problem provided asks to find the volume of a solid obtained by rotating the region under the curve from to about the y-axis. This requires applying the method of cylindrical shells.
Step-by-step solution:
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Formula for volume using cylindrical shells:
When rotating around the y-axis, the volume is given by the formula: where is the function describing the curve, and and are the limits of integration (in this case, and ).
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Identify the function and limits:
- The limits are from to .
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Set up the integral:
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Substitute :
- When , .
- When , .
The integral becomes:
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Integrate:
So, the volume of the solid is cubic units.
Would you like further details or have any questions?
Here are some related problems:
- What is the volume when rotating the same region around the x-axis?
- How would you solve this problem if the curve were rotated around the line ?
- Can you calculate the surface area of the solid obtained by this rotation?
- How does the volume change if the curve is rotated from to ?
- What is the interpretation of the integral when rotating a curve about the y-axis?
Tip: When using the cylindrical shells method, always ensure that you set up the integral correctly in terms of the radius and height of the shell.
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Math Problem Analysis
Mathematical Concepts
Calculus
Volume of Solids of Revolution
Cylindrical Shells
Formulas
V = 2π∫(x * f(x)) dx (for cylindrical shells)
u-substitution: u = 4 - x², du = -2x dx
Theorems
Method of Cylindrical Shells
u-Substitution
Suitable Grade Level
College Level (Calculus II)
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