Volumes by Disks and Washers
Revolve an area around an axis.
Key idea
Revolve an area around an axis.
In depth
Slice the solid perpendicular to the axis of revolution. Each slice is a disk of area πr² or a washer π(R² − r²). Integrate the cross-sectional areas.
Worked example
y = √x on [0, 4] rotated about the x-axis: V = π∫₀⁴ x dx = 8π.
Formula
Real-world uses
- Calculating the volume of a rounded bottle, a lathe-turned part or a vase
Common mistake
Using the diameter or forgetting to square the radius.
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 Disks and Washers?
Revolve an area around an axis. It is a topic at the expert level in Calculus.
What is the formula for Volumes by Disks and Washers?
The key formula is V = π∫ r² dx.
Where is Volumes by Disks and Washers used in real life?
Real-world examples: Calculating the volume of a rounded bottle, a lathe-turned part or a vase.
What should I learn before Volumes by Disks and Washers?
Make sure you are comfortable with Area Between Curves, the previous topic in Calculus.
What is a common mistake with Volumes by Disks and Washers?
Using the diameter or forgetting to square the radius.
How can I practise Volumes by Disks and Washers?
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 Disks and Washers?
Move on to Volumes by Shells 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.