Effortlessly calculate the volume of a solid of revolution using the Disk Method. This tool handles polynomial functions of the form f(x) = cxⁿ rotated around the x-axis.
Volume Solid Revolution Calculator
Volume Solid Revolution Calculator Formula
For f(x) = cxⁿ:
V = π ∫ (cxⁿ)² dx = π ∫ c²x²ⁿ dx
V = π [ (c² * x²ⁿ⁺¹) / (2n + 1) ] from a to b
Formula Sources: Wolfram MathWorld, OpenStax Calculus
Variables:
- Coefficient (c): The multiplier of the variable x in your function.
- Power (n): The exponent to which the variable x is raised.
- Lower Bound (a): The starting x-coordinate of the region.
- Upper Bound (b): The ending x-coordinate of the region.
What is Volume Solid Revolution Calculator?
The Volume Solid Revolution Calculator is a specialized mathematical tool designed to compute the volume of a 3D shape generated by rotating a 2D curve around a fixed axis (usually the x-axis or y-axis). In calculus, this is a fundamental application of integration.
This process, often called the “Disk Method” or “Washer Method,” slices the solid into infinitely thin circular disks. By summing the volumes of these disks using a definite integral, we can find the exact volume of complex shapes like cones, spheres, and paraboloids that are difficult to measure with basic geometry.
How to Calculate Volume Solid Revolution (Example)
- Identify the function: Let’s use f(x) = x² (c=1, n=2).
- Set the bounds: We want to rotate the area from x = 0 to x = 2.
- Set up the integral: V = π ∫₀² (x²)² dx = π ∫₀² x⁴ dx.
- Integrate: The antiderivative of x⁴ is (1/5)x⁵.
- Evaluate: π [ (1/5)(2)⁵ – (1/5)(0)⁵ ] = π [ 32/5 ] = 6.4π.
- Final Result: Approximately 20.106.
Related Calculators
Explore more geometry and calculus tools:
- Surface Area of Revolution Calculator
- Definite Integral Solver
- Cylindrical Shell Method Calculator
- Centroid of Planar Region Tool
Frequently Asked Questions (FAQ)
What is the difference between the Disk and Shell methods? The Disk method slices the solid perpendicular to the axis of revolution, while the Shell method slices it parallel to the axis using concentric cylinders.
Can this calculator handle rotation around the y-axis? This specific module is optimized for x-axis rotation. For y-axis rotation, you would typically express the function as x = f(y) and integrate with respect to y.
Why is π included in the formula? Since the cross-section of a solid of revolution is always a circle, the area of each slice is πr², where the radius r is the value of the function f(x).
What happens if the function goes below the x-axis? Because the radius is squared in the formula [f(x)]², the result remains positive, correctly representing volume.