Volumes by Shells
Cylindrical shell method.
Key idea
Cylindrical shell method.
In depth
Use cylindrical shells parallel to the axis of rotation: V = 2π ∫ (radius)(height) dx. It is often easier when slicing gives awkward washers.
Worked example
y = x² on [0, 2] rotated about the y-axis: V = 2π∫ x · x² dx = 8π.
Formula
Real-world uses
- Computing the volume of a bowl or a dome made by spinning a curve around an axis
Common mistake
Using the wrong radius, which is the distance to the axis of rotation.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Volumes by Shells?
Cylindrical shell method. It is a topic at the expert level in Calculus.
What is the formula for Volumes by Shells?
The key formula is V = 2π∫ r h dx.
Where is Volumes by Shells used in real life?
Real-world examples: Computing the volume of a bowl or a dome made by spinning a curve around an axis.
What should I learn before Volumes by Shells?
Make sure you are comfortable with Volumes by Disks and Washers, the previous topic in Calculus.
What is a common mistake with Volumes by Shells?
Using the wrong radius, which is the distance to the axis of rotation.
How can I practise Volumes by Shells?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Volumes by Shells?
Move on to Arc Length next.
Related expert topics in Calculus
- Derivatives of Inverse Trig – Derivatives of arcsin, arccos, arctan.
- Implicit Differentiation – Differentiate both sides of F(x,y)=0 treating y as a function of x.
- Logarithmic Differentiation – Take logs before differentiating products and powers.
- Related Rates – Link rates of change of connected quantities.
- Linear Approximation – Approximate f near a using its tangent line.
- Differentials – Use dy = f′(x)dx for small changes.