Volumes of Revolution

A study of how a curve, swept about an axis, becomes a solid.

Curve

Domain

start, a0.00

end, b4.00

Axis of revolution

Display

Riemann disks

n disks24

wireframe overlay coordinate axes

Volume — disk method

V = π ∫ [f(x)]² dx

0.000 approx.

Drag to orbit. Right-click drag (or shift+drag) to pan. Scroll to zoom. Hit Reset view to re-center on the solid. The disks visualise Riemann's idea: the volume is the limit of stacked thin cylinders as their thickness approaches zero.

Plate I. The solid generated by revolving y = √x about the x-axis.

θ ∈ [0, 2π]

n → ∞

generating curve

cross-section disk

axis of revolution

θangle of revolution

0°90°180°270°360°