Volume Of Solid Of Revolution Calculator

Volume of Solid of Revolution Formula:

\[ V = 2 \times \pi \times A_{Curve} \times r_{Area Centroid} \]

Volume of Solid of Revolution:

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1. What is the Volume of Solid of Revolution?

The Volume of Solid of Revolution is the total quantity of three dimensional space enclosed by the entire surface of the Solid of Revolution formed by rotating a plane curve around a fixed axis.

2. How Does the Calculator Work?

The calculator uses the Volume of Solid of Revolution formula:

\[ V = 2 \times \pi \times A_{Curve} \times r_{Area Centroid} \]

Where:

\( V \) — Volume of Solid of Revolution (m³)
\( \pi \) — Archimedes' constant (≈3.141592653589793)
\( A_{Curve} \) — Area under Curve Solid of Revolution (m²)
\( r_{Area Centroid} \) — Radius at Area Centroid of Solid of Revolution (m)

Explanation: This formula calculates the volume generated when a plane area revolves around an external axis, using the Pappus centroid theorem.

3. Importance of Volume Calculation

Details: Accurate volume calculation is crucial for engineering applications, architectural design, manufacturing processes, and various scientific computations involving three-dimensional objects created by revolution.

4. Using the Calculator

Tips: Enter the area under the curve in square meters and the radius at the area centroid in meters. Both values must be positive numbers.

5. Frequently Asked Questions (FAQ)

Q1: What is a Solid of Revolution?
A: A Solid of Revolution is a three-dimensional object obtained by rotating a two-dimensional curve or area around a straight line (the axis of revolution).

Q2: When is this formula applicable?
A: This formula applies specifically when using the theorem of Pappus for finding volumes of solids of revolution, requiring the area and the distance traveled by its centroid.

Q3: What are common applications of solids of revolution?
A: Common applications include calculating volumes of tanks, vessels, architectural domes, machine parts, and various manufactured components with rotational symmetry.

Q4: Are there limitations to this method?
A: This method requires knowing the centroid position of the area being revolved. For complex shapes, alternative methods like disk/washer or shell methods may be needed.

Q5: How accurate is this calculation?
A: The calculation is mathematically exact for the given inputs, assuming the centroid position is correctly determined and the area is accurately measured.