Math Problem Statement
Given the function f(x)=3x with the domain x=[0, 2], find the volume of a surface bounded by the graph of f, the axises and the line y = 6, if it was dragged 360 degrees around the x-axis.
Solution
To solve this problem, we are tasked with finding the volume of a solid formed by rotating the region bounded by the graph of the function , the x-axis, and the line around the x-axis. The domain of is given as .
We will use the method of cylindrical shells to calculate the volume. The general formula for the volume of a solid formed by rotating a region around the x-axis using cylindrical shells is:
Step 1: Define the radius and height of the cylindrical shells
- Radius: The distance from the x-axis to the function is simply , which gives the radius of each cylindrical shell.
- Height: The height of each cylindrical shell is the difference between the line and the graph of . So the height of each shell is .
Step 2: Set up the integral
Now, we set up the integral to find the volume:
Step 3: Simplify the integrand
Simplify the expression inside the integral:
Step 4: Integrate
Now, perform the integration:
Thus, the volume integral becomes:
Step 5: Evaluate the integral
Now, evaluate the integral at the limits and :
Final Answer:
The volume of the solid is cubic units.
Would you like a more detailed explanation of any step, or do you have further questions? Here's a list of related topics you might want to explore:
- Understanding the method of cylindrical shells in more detail.
- How to visualize the solid formed by rotation.
- The disc method versus the cylindrical shell method for finding volumes.
- Volumes of solids of revolution for different curves and limits.
- How to calculate the volume for rotations around other axes.
Tip: When rotating around the x-axis or y-axis, always start by carefully defining the radius and height of the cylindrical shells to simplify your integration!
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Math Problem Analysis
Mathematical Concepts
Calculus
Volume of Solids of Revolution
Cylindrical Shells
Formulas
V = 2π ∫[a, b] (radius × height) dx
Theorems
Method of Cylindrical Shells
Suitable Grade Level
Grades 11-12
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