Volumes of Revolution: Cylindrical Shells: Fresh Take

Lumen Learning

Volumes of Revolution: Cylindrical Shells: Fresh Take

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 $y$ -axis:
  - $V = \int_a^b 2\pi x f(x) dx$
- Key formula for rotation around $x$ -axis:
  - $V = \int_c^d 2\pi y g(y) dy$
Adaptability:
- Can be modified for rotation around lines other than coordinate axes
- For rotation around $x = -k$ : $V = \int_a^b 2\pi (x+k) f(x) dx$
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 $y$ -axis rotations of $x$ -defined functions
- Useful when other methods require multiple integrals

Key Concepts

Shell Volume:
- Approximated by $V_{shell} \approx 2\pi x^* f(x^*) \Delta x$ for y-axis rotation
- Height of shell is the function value, radius is the $x$ -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 $f(x) - g(x)$ for shell height
Method Selection:
- Consider the axis of rotation and how functions are defined
- Choose method that results in simplest integration

Define $R$ as the region bounded above by the graph of $f(x)={x}^{2}$ and below by the $x$ -axis over the interval $\left[1,2\right].$ Find the volume of the solid of revolution formed by revolving $R$ around the $y\text{-axis}.$

$\frac{15\pi }{2}$ units³

Define $R$ as the region bounded above by the graph of $f(x)=3x-{x}^{2}$ and below by the $x\text{-axis}$ over the interval $\left[0,2\right].$ Find the volume of the solid of revolution formed by revolving $R$ around the $y\text{-axis}.$

$8\pi$ units³

Define $Q$ as the region bounded on the right by the graph of $g(y)=3\text{/}y$ and on the left by the $y\text{-axis}$ for $y\in \left[1,3\right].$ Find the volume of the solid of revolution formed by revolving $Q$ around the $x\text{-axis}.$

$12\pi$ units³

Watch the following video to see the worked solution to this example.

For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.

You can view the transcript for this segmented clip of “2.3 Volumes of Revolution: Cylindrical Shells” here (opens in new window).

Define $R$ as the region bounded above by the graph of $f(x)={x}^{2}$ and below by the $x\text{-axis}$ over the interval $\left[0,1\right].$ Find the volume of the solid of revolution formed by revolving $R$ around the line $x=-2.$

$\frac{11\pi }{6}$ units³

Watch the following video to see the worked solution to this example.

For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.

You can view the transcript for this segmented clip of “2.3 Volumes of Revolution: Cylindrical Shells” here (opens in new window).

Define $R$ as the region bounded above by the graph of $f(x)=x$ and below by the graph of $g(x)={x}^{2}$ over the interval $\left[0,1\right].$ Find the volume of the solid of revolution formed by revolving $R$ around the $y\text{-axis}.$

$\frac{\pi }{6}$ units³