Solid of Revolution Math

Volume = ---
0.0 4.0
60

What is a Solid of Revolution?

Quick Answer: A 3D shape formed by rotating a 2D curve around an axis. This tool visualises the solid generated when y = f(x) is revolved around the x‑axis (or y‑axis), and computes its volume using the disc method: V = π ∫ f(x)² dx.

Theory of the Disc Method

When a curve y = f(x) from x = a to x = b is revolved around the x‑axis, each cross‑section is a disc of radius f(x). The volume is the integral of the area of these discs: V = π ∫ab [f(x)]2 dx. For revolution around the y‑axis using x = g(y), the formula is similar. This tool evaluates f(x) with Math.js, constructs a 3D mesh using Three.js via LatheGeometry, and approximates the volume numerically. You can rotate, zoom, and pan to explore the solid from any angle.

Step-by-Step Examples

Example 1: Cone from y = x

  1. Enter f(x) = x, set a = 0, b = 4. Revolve around X‑axis.
  2. You'll see a cone with vertex at the origin. Volume = π ∫ x² dx = π (4³/3) ≈ 67.02.
  3. Toggle to Y‑axis revolution to see a different solid (a cylinder with a conical hole).

Example 2: Sphere from y = √(r² - x²)

  1. Use f(x) = sqrt(9 - x^2), a = -3, b = 3. Revolve around X‑axis.
  2. The upper half of the circle generates a full sphere of radius 3. Volume = 4/3 π r³ ≈ 113.1.

Example 3: Wine Glass Shape

  1. Set f(x) = 0.5 + 0.2*sin(2*x), a = 0, b = 6.28 (2π). Revolve around X‑axis.
  2. You'll see a rippled solid resembling a vase or wine glass.

Frequently Asked Questions

How do I enter the function?

Type any valid expression using x, numbers, +, -, *, /, ^, sin, cos, sqrt, exp, log, abs, etc. Example: sqrt(x) + 1.

What does the solid represent?

It's the 3D shape created by rotating the area under the curve y = f(x) from a to b around the chosen axis.

How is the volume calculated?

The tool approximates the integral of π f(x)² dx using the same points that build the mesh. The exact value is shown in the overlay.

Can I rotate the view?

Yes, drag with the mouse to rotate, scroll to zoom, right‑click to pan. Reset Camera brings back the default view.

What does "Segments" control?

It sets the number of subdivisions along the curve. Higher values give a smoother solid but may slow down older devices.