Revolve — Solids of revolution

SOLIDS OF REVOLUTION

One region. A new dimension.

About y = 0

The 2D region

BEFORE

The rotated solid

0°

Drag to orbit · Scroll to zoom

Rotation0°

Explore a slice

Direct setup: washers

Move piece x = 0.5dx

Number of pieces N = 24Δx = 0.041667

Drag in the 2D plot to move the piece. In 3D, choose “Move piece” above. More pieces means less thickness.

A vertical strip becomes a washer. Its thickness is dx.

Build the whole solid

Add successive pieces, then make them thinner.

0 / 24 pieces

Pieces added0 / 24

Cutaway viewThe purple outline is the target solid. Gold marks the newest piece.

From geometry to the integral

WASHER METHOD

Each visible piece has finite thickness Δx, Δy, or Δρ. Increasing N makes the pieces thinner; the finite sums approach the volume integral. A cutaway changes only the view, not the full-revolution volumes.

Volume of the solid

Reveal the answer when you’re ready.

Washer: perpendicular to the axis. Shell: parallel to the axis. The region stays the same.