Find the Volume of the Solid Generated

Volume of Revolution Formula:

\[ V = \int_{a}^{b} \pi \times [f(x)]^2 \,dx \]

Volume (cubic units):

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1. What is Volume of Revolution?

Definition: The volume generated by rotating a function around the x-axis between two points.

Purpose: Used in mathematics and engineering to calculate volumes of complex shapes defined by functions.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ V = \int_{a}^{b} \pi \times [f(x)]^2 \,dx \]

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 function is squared, multiplied by π, and integrated between the limits to find the volume.

3. Importance of Volume Calculation

Details: Essential for engineering designs, manufacturing, and any application requiring precise volume calculations of rotational solids.

4. Using the Calculator

Tips: Enter the function (e.g., "x^2", "sin(x)"), the lower and upper limits. The upper limit must be greater than the lower limit.

5. Frequently Asked Questions (FAQ)

Q1: What functions can I use?
A: The calculator supports basic functions (x^2, sin(x), etc.). For complex functions, specialized software may be needed.

Q2: What if my upper limit is less than lower limit?
A: The calculator will return an empty result. Limits must be a ≤ b.

Q3: Can I rotate around the y-axis instead?
A: This calculator is for x-axis rotation. For y-axis, the formula would need adjustment.

Q4: How accurate are the results?
A: Accuracy depends on the function complexity. Simple polynomials give exact results.

Q5: What units does this use?
A: The units depend on your input function. Results are in cubic units of whatever your x-axis represents.