Rotating Volume Calculator Online

The Rotating Volume Calculator helps in visualizing and calculating the volume of solids of revolution. These solids are formed when a plane area is rotated around a line (axis) next to the area, forming a three-dimensional solid.

Formula of Rotating Volume Calculator

The volume of solids of revolution is calculated using integral calculus. Here are the calculations based on the axis of rotation:

Rotation about the X-Axis: For a shape defined by the curve y = f(x), from x = a to x = b, the volume V is calculated by: V = integral from a to b of pi [f(x)]^2 dx Here, pi represents the constant pi, and [f(x)]^2 is the squared distance from the curve to the x-axis, integral across the bounds of a to b.

Rotation about the Y-Axis: For a shape where x = g(y) from y = c to y = d, the volume V is calculated as: V = integral from c to d of pi [g(y)]^2 dy This formula also involves integrating the squared distance from the curve to the y-axis over the interval from y = c to y = d.

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Example of Rotating Volume Calculator

et's go through an example of using the Rotating Volume Calculator with a semicircular region. Suppose we have a semicircle defined by the equation y = sqrt(r^2 - x^2), where r is the radius of the semicircle. We want to calculate the volume of the solid formed when this semicircle is rotated around the x-axis.

Step-by-Step Calculation:

1. Identify the Shape and Equation:
   • The shape is a semicircle.
   • The equation defining this shape is y = sqrt(r^2 - x^2).
2. Determine the Axis of Rotation:
   • The semicircle is rotate around the x-axis.
3. Set Up the Volume Formula:
   • The formula to calculate the volume when rotating around the x-axis is:
     • V = integral from -r to r of pi [sqrt(r^2 - x^2)]^2 dx
4. Simplify the Equation:
   • The integral simplifies to pi times the integral from -r to r of (r^2 - x^2) dx.
5. Calculate the Integral:
   • The integral of (r^2 - x^2) from -r to r equals 2/3 pi r^3. This calculation can be perform by splitting the integral into r^2x - x^3/3 and evaluating from -r to r.
6. Compute the Volume:
   • Substitute the result of the integral into the volume formula:
     • V = pi * 2/3 pi r^3 = 4/3 pi r^3

Conclusion:

The volume of the solid formed by rotating the given semicircle around the x-axis is 4/3 pi r^3. This example illustrates how to use the Rotating Volume Calculator to simplify the process of computing volumes, making it a practical tool for educational and professional applications.

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