- Determine the volume of a solid formed by rotating a region around an axis using cylindrical shells
- Evaluate the benefits and limitations of different methods (disk, washer, cylindrical shells) for calculating volumes
Cylindrical Shells Method
The Main Idea
- Cylindrical Shells Method:
- Used for solids of revolution, especially when integrating parallel to the axis of rotation
- Key formula for rotation around [latex]y[/latex]-axis:
- [latex]V = \int_a^b 2\pi x f(x) dx[/latex]
- Key formula for rotation around [latex]x[/latex]-axis:
- [latex]V = \int_c^d 2\pi y g(y) dy[/latex]
- Adaptability:
- Can be modified for rotation around lines other than coordinate axes
- For rotation around [latex]x = -k[/latex]: [latex]V = \int_a^b 2\pi (x+k) f(x) dx[/latex]
- Comparison with Other Methods:
- Disk Method: Used when rotating around an axis perpendicular to the rectangles
- Washer Method: Similar to disk method, but for solids with cavities
- Choice depends on ease of integration and problem setup
- Advantages:
- Often simplifies calculations for [latex]y[/latex]-axis rotations of [latex]x[/latex]-defined functions
- Useful when other methods require multiple integrals
Key Concepts
- Shell Volume:
- Approximated by [latex]V_{shell} \approx 2\pi x^* f(x^*) \Delta x[/latex] for y-axis rotation
- Height of shell is the function value, radius is the [latex]x[/latex]-coordinate
- Axis of Rotation:
- Formula adjusts based on the axis of rotation
- Radius term in integral changes for different axes
- Multiple Functions:
- For regions bounded by two functions, use [latex]f(x) - g(x)[/latex] for shell height
- Method Selection:
- Consider the axis of rotation and how functions are defined
- Choose method that results in simplest integration
Define [latex]R[/latex] as the region bounded above by the graph of [latex]f(x)={x}^{2}[/latex] and below by the [latex]x[/latex]-axis over the interval [latex]\left[1,2\right].[/latex] Find the volume of the solid of revolution formed by revolving [latex]R[/latex] around the [latex]y\text{-axis}.[/latex]
Define [latex]R[/latex] as the region bounded above by the graph of [latex]f(x)=3x-{x}^{2}[/latex] and below by the [latex]x\text{-axis}[/latex] over the interval [latex]\left[0,2\right].[/latex] Find the volume of the solid of revolution formed by revolving [latex]R[/latex] around the [latex]y\text{-axis}.[/latex]
Define [latex]Q[/latex] as the region bounded on the right by the graph of [latex]g(y)=3\text{/}y[/latex] and on the left by the [latex]y\text{-axis}[/latex] for [latex]y\in \left[1,3\right].[/latex] Find the volume of the solid of revolution formed by revolving [latex]Q[/latex] around the [latex]x\text{-axis}.[/latex]
Define [latex]R[/latex] as the region bounded above by the graph of [latex]f(x)={x}^{2}[/latex] and below by the [latex]x\text{-axis}[/latex] over the interval [latex]\left[0,1\right].[/latex] Find the volume of the solid of revolution formed by revolving [latex]R[/latex] around the line [latex]x=-2.[/latex]
Define [latex]R[/latex] as the region bounded above by the graph of [latex]f(x)=x[/latex] and below by the graph of [latex]g(x)={x}^{2}[/latex] over the interval [latex]\left[0,1\right].[/latex] Find the volume of the solid of revolution formed by revolving [latex]R[/latex] around the [latex]y\text{-axis}.[/latex]