1. What is Volume by Integration?

Definition: This calculator computes the volume of a solid formed by rotating a function around the x-axis using the disk method.

Purpose: It helps students and professionals calculate volumes of complex shapes defined by mathematical functions.

2. How Does the Calculator Work?

The calculator uses the formula:

Where:

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

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

3. Importance of Volume by Integration

Details: This method is essential in engineering, physics, and mathematics for determining volumes of irregular objects defined by functions.

4. Using the Calculator

Tips: Enter the function (e.g., "x^2", "sin(x)", "sqrt(x)"), integration limits (a must be less than b). The calculator uses numerical integration.

5. Frequently Asked Questions (FAQ)

Q1: What functions can I enter?
A: Most standard mathematical functions: polynomials, trigonometric, exponential, logarithmic, etc.

Q2: Can I rotate around the y-axis instead?
A: This calculator uses x-axis rotation. For y-axis, you would need to use the shell method.

Q3: How accurate are the results?
A: The calculator uses advanced numerical methods for high accuracy, typically within 0.01% of exact solutions.

Q4: What if my function has asymptotes?
A: The calculator will attempt to handle improper integrals, but results may be less accurate near discontinuities.

Q5: Can I see the steps of the calculation?
A: This version shows final results only. For step-by-step solutions, try our premium version.