Volume of Solids of Revolution Solver | Disk, Washer & Shell Calculator

SOLIDS OF REVOLUTION SOLVER

Disk, Washer & Cylindrical Shell Methods

Calculus & 3D Integration

Select Geometry Scenario: Paraboloid: f(x) = x^2, g(x) = 0 around x-axis on [0, 2] Washer: f(x) = x, g(x) = x^2 around x-axis on [0, 1] Shell Method: f(x) = x^2, g(x) = 0 around y-axis on [0, 2] Shifted Axis: f(x) = sqrt(x), g(x) = 0 around y = -1 on [0, 4] Unit Hemisphere: f(x) = sqrt(1 - x^2), g(x) = 0 around x-axis on [0, 1]

Outer Function $f(x)$:

Inner Function $g(x)$ (0 for Disk):

Lower Bound $a$:

Upper Bound $b$:

Axis Orientation: Horizontal Axis (y = c) Vertical Axis (x = c)

Axis Constant ($c$):

3D Solid Volume Analysis

Recommended Method Disk / Washer

Calculated Volume ($V$) 20.1062

Exact Volume Multiplier ($\pi$-notation) 6.4000 π

Step-by-Step Integral Derivation

Volume of Solids of Revolution: Disk, Washer & Cylindrical Shell Methods

Calculating the volume of a solid of revolution involves rotating a two-dimensional region bounded by functions around a specified line (the axis of revolution) in three-dimensional space. Selecting between the Disk, Washer, and Cylindrical Shell methods depends on the axis orientation and the variable of integration.

1. Summary of Rotational Methods

Method	Axis of Revolution	Slice Geometry	Integral Formula
Disk Method	Parallel to slice axis ($y = c$)	Solid circular cross-section	$$V = \pi \int_{a}^{b} [R(x)]^2 \, dx$$
Washer Method	Parallel to slice axis ($y = c$)	Circular cross-section with hole	$$V = \pi \int_{a}^{b} \left( [R(x)]^2 - [r(x)]^2 \right) \, dx$$
Cylindrical Shell Method	Perpendicular to slice axis ($x = c$)	Thin concentric cylindrical shell	$$V = 2\pi \int_{a}^{b} r(x) h(x) \, dx$$

2. Choosing Between Washer and Shell Methods

• Washer Method (dx): Best when rotating around horizontal lines ($y = c$) and function curves are naturally given as $y = f(x)$. Slices are perpendicular to the axis of rotation.
• Cylindrical Shell Method (dx): Ideal when rotating around vertical lines ($x = c$) without needing to invert $y = f(x)$ into $x = f^{-1}(y)$. Slices are parallel to the axis of rotation.

Frequently Asked Questions

What happens when the axis of rotation is shifted?

When rotating around $y = c$ or $x = c$ instead of the coordinate axes ($y = 0$ or $x = 0$), adjust the radius expressions accordingly: $R(x) = |f(x) - c|$ or $r(x) = |x - c|$.

Can a volume generated by revolution be negative?

No. Absolute values or squared radius terms ensure $R(x) \ge r(x) \ge 0$, rendering volume results non-negative.

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