1. What is Volume by Integration?

Definition: This calculator estimates the volume of a solid of revolution using the disk method of integration.

Purpose: It helps mathematicians, engineers, and students determine volumes of complex shapes created by rotating a function around an axis.

2. How Does the Calculator Work?

The calculator uses the disk method formula:

Where:

• \( V \) — Volume of the solid (cubic units)
• \( f(x) \) — Function being rotated
• \( a \) — Lower limit of integration
• \( b \) — Upper limit of integration

Explanation: The method sums up infinitely thin circular disks along the axis of rotation to calculate the total volume.

3. Importance of Volume by Integration

Details: This method allows calculation of volumes for shapes that cannot be easily measured using standard geometric formulas.

4. Using the Calculator

Tips: Enter the function f(x) in terms of x (e.g., "x^2", "sin(x)", "sqrt(x)"), the lower limit a, and upper limit b (a < b).

5. Frequently Asked Questions (FAQ)

Q1: What functions can I enter?
A: The calculator supports standard mathematical functions - polynomials, trigonometric, exponential, etc.

Q2: What if my function has vertical asymptotes?
A: The integral may not converge. Choose limits that avoid singularities.

Q3: Can I use this for other methods like shell method?
A: This calculator specifically implements the disk method.

Q4: How accurate are the results?
A: Accuracy depends on the numerical integration technique used.

Q5: What units should I use?
A: Use consistent units - the result will be in cubic units of your input.