1. What is a Volume by Integration Calculator?

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

Purpose: It helps students and professionals calculate volumes of complex shapes generated by rotating a function around the x-axis.

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 circular disks along the axis of rotation to calculate the total volume.

3. Importance of Volume Calculation

Details: Accurate volume calculations are essential in engineering, physics, and manufacturing for designing objects and calculating material requirements.

4. Using the Calculator

Tips: Enter the function to be rotated (using standard mathematical notation), the integration limits (a and b), and ensure b > a.

5. Frequently Asked Questions (FAQ)

Q1: What types of functions can I enter?
A: The calculator supports polynomial, trigonometric, exponential, and logarithmic functions (implementation dependent).

Q2: What if my function has discontinuities?
A: The function must be continuous and defined over [a,b]. Results may be inaccurate for discontinuous functions.

Q3: Can I use this for the washer method?
A: This implements the disk method. For washers, you would need to modify the integrand to (f(x)^2 - g(x)^2).

Q4: How accurate are the results?
A: Accuracy depends on the numerical integration method used (typically very accurate for well-behaved functions).

Q5: What about rotation around the y-axis?
A: For y-axis rotation, you would need to use the shell method or express x as a function of y.