Math Problem Statement
Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line.
y = x2, y = 4x;
about the y-axis
Solution
To find the volume of the solid obtained by rotating the region bounded by the curves and about the -axis, we use the method of cylindrical shells. Here are the steps involved:
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Determine the points of intersection: To find the bounds of integration, we first solve for the points where the curves and intersect.
Set : So, and .
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Set up the integral: When using the shell method, the volume is given by: where is the outer function and is the inner function in terms of . Here, and .
The bounds are to .
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Integrate: Simplify the integrand:
Now, integrate term-by-term:
Calculate the integrals:
Thus, the volume of the solid obtained by rotating the region bounded by and about the -axis is .
Would you like more details on any step or have any questions?
Here are some related questions for further exploration:
- How would the volume change if the region were rotated about the x-axis instead?
- What if the region were rotated about the line ?
- How can you use the disk method for this problem?
- What is the volume if the region is rotated about the line ?
- Can you find the centroid of the region bounded by the curves?
- How does the method of washers differ from the method of shells?
- What if the region were bounded by and ?
- How can numerical methods approximate the volume of more complex regions?
Tip: Always check the points of intersection and the setup of the integral carefully to avoid errors in bounds or integrand simplification.
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Math Problem Analysis
Mathematical Concepts
Integral calculus
Volume of revolution
Cylindrical shells method
Formulas
Volume using cylindrical shells
Theorems
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Suitable Grade Level
University level
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