Volume of Revolution Calculator

Function f(x)

Method

Lower Bound:

Upper Bound:

Axis of Revolution

Understanding Volumes of Revolution

Integration Methods

Disk Method

V = π∫[a to b] [f(x)]² dx

For rotation around x-axis
Cross sections are circles
Radius = f(x)

Washer Method

V = π∫[a to b] ([f(x)]² - [g(x)]²) dx

For hollow solids
Outer radius = f(x)
Inner radius = g(x)

Shell Method

V = 2π∫[a to b] x·f(x) dx

For rotation around y-axis
Radius = x
Height = f(x)

Applications and Examples

Common Shapes

Sphere: y = √(r² - x²), rotated around x-axis
Cone: y = mx, rotated around x-axis
Torus: Circle rotated around y-axis

Real-world Applications

Engineering Design
Fluid Containers
Architectural Elements
Manufacturing Processes