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Expertly reviewed by David Chen, MS in Mathematics. This tool follows standard calculus principles for volume integration.

Easily calculate the volume of a solid of revolution using the Disk Method. Enter your function parameters and integration limits to find the exact volume rotated around the x-axis.

Volume Revolution Calculator

Calculated Volume 0.00 Cubic Units

Volume Revolution Calculator Formula

V = π ∫ab [f(x)]² dx

Reference: Wolfram MathWorld – Solid of Revolution

Variables:

  • f(x): The function representing the curve to be revolved.
  • k: The coefficient or constant value of the function.
  • a: The starting point on the x-axis (lower limit of integration).
  • b: The ending point on the x-axis (upper limit of integration).
  • π (Pi): Mathematical constant (~3.14159).

What is a Volume Revolution Calculator?

A volume revolution calculator is a specialized tool used in calculus to determine the volume of a three-dimensional solid formed by rotating a two-dimensional curve around an axis (usually the x-axis or y-axis). This method is fundamental in engineering and physics for designing parts like pistons, bottles, and architectural domes.

By using the Disk Method, we sum up an infinite number of thin cylindrical disks with radius f(x) and thickness dx. The integral accumulates these areas to provide the total volume.

How to Calculate Volume of Revolution (Example)

Let’s calculate the volume of f(x) = 2x from x=0 to x=3 rotated around the x-axis:

  1. Identify the function: f(x) = 2x, so [f(x)]² = 4x².
  2. Set up the integral: V = π ∫₀³ (4x²) dx.
  3. Find the antiderivative: 4x³/3.
  4. Apply limits: [4(3)³/3] – [4(0)³/3] = 4(9) – 0 = 36.
  5. Multiply by π: V = 36π ≈ 113.10.

Related Calculators

• Surface Area of Revolution Calculator
• Definite Integral Calculator
• Centroid of a Solid Calculator
• Cylinder Volume Calculator

Frequently Asked Questions (FAQ)

What is the difference between the Disk and Washer methods?

The disk method is used when the area being rotated is flush against the axis of revolution. The washer method is used when there is a gap (a hole) between the curve and the axis.

Can I rotate around the y-axis?

Yes, but the formula changes to V = π ∫ [f(y)]² dy, and the limits must be on the y-axis.

Why is Pi (π) in the formula?

Because the cross-section of a solid of revolution is always a circle, and the area of a circle is πr².

What are the units of the result?

Since it is volume, the result is in cubic units (e.g., cm³, in³).

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