1. What is the Volume of Rotation?

The volume of rotation is the volume of a three-dimensional object obtained by rotating a two-dimensional curve around an axis. The disk method calculates this volume by summing up infinitely thin circular disks along the axis of rotation.

2. How Does the Calculator Work?

The calculator uses the disk method formula:

Where:

• \( R(x) \) — Radius function (distance from curve to axis)
• \( a \) — Lower limit of integration
• \( b \) — Upper limit of integration
• \( dx \) — Infinitesimal width along x-axis

Explanation: The calculator performs numerical integration by approximating the integral as a Riemann sum with many small steps.

3. Importance of Volume Calculation

Details: Calculating volumes of rotation is essential in engineering, physics, and manufacturing for determining capacities, material requirements, and structural properties of rotational objects.

4. Using the Calculator

Tips: Enter the radius function (e.g., "x^2" or "sin(x)"), the integration limits (a and b), and optionally adjust the number of steps for more precise calculations.

5. Frequently Asked Questions (FAQ)

Q1: What functions can I enter?
A: Basic algebraic (x^2, 2*x+1) and trigonometric (sin(x), cos(x)) functions are supported. For complex functions, use proper mathematical notation.

Q2: How accurate is the calculation?
A: Accuracy depends on the number of steps. More steps = more accurate but slower calculation.

Q3: Can I rotate around the y-axis?
A: This calculator assumes rotation around the x-axis. For y-axis rotation, you would need to express x as a function of y.

Q4: What if my function crosses the axis?
A: The calculator will still work but may give negative volumes if the function goes below the axis. Absolute value may be needed.

Q5: Can I use this for real-world applications?
A: For critical applications, verify results with exact integration or professional software.