Volume of Solid of Revolution Calculator
Compute volume of a revolution solid. Enter curves, rotation line, and method; examples, live steps, and results update instantly today.
Equation Preview
Helping Notes
Fields match standard practice: curves, rotation line, and method. Limits are found from intersections automatically.
Use functions of x such as sqrt(x), x^2+1, sin(x). If needed, adjust inputs so intersections exist.
Results
Volume V
Core Integral
Interval
What Is a Volume of Solid of Revolution Calculator?
A Volume of Solid of Revolution Calculator computes the volume generated when a 2D region is rotated around a line (axis). It sets up the correct integral using the disk, washer, or cylindrical shell method, then evaluates symbolically (when possible) and numerically. You provide bounding curves, limits, and the rotation axis—about the x‑axis, y‑axis, or a shifted line such as or . The tool determines radii or shell dimensions, builds the integrand, and returns exact and approximate volumes with clear steps. This makes it ideal for calculus homework, engineering sketches, and quick checks before submitting solutions.
About the Volume of Solid of Revolution Calculator
The calculator chooses an orientation to simplify setup: washers use slices perpendicular to the axis; shells use slices parallel to the axis. For rotation about the x‑axis (or line ) with functions of , radii are vertical distances. For rotation about the y‑axis (or line ) with functions of , radii are horizontal distances. If curves cross, the interval is split and outer/inner roles are reassessed. The engine also handles diameters, absolute values for shifted axes, and can approximate using Riemann sums when symbolic primitives are not available. Units are carried through, so inputs in centimeters yield cubic centimeters, with optional conversions to liters or gallons.
Disk method (no hole):
Washer method (hole):
About the y‑axis (or x=c):
Shell method (parallel slices): , or
Shifted axis distances:
Pappus check (optional): , where is region area and is centroid distance to the axis.
How to Use This Volume of Solid of Revolution Calculator
- Enter the bounding curves and choose your rotation axis (x‑axis, y‑axis, or a line like y = c or x = c).
- Select method: washers/disks (perpendicular slices) or shells (parallel slices). The tool suggests the simpler orientation automatically.
- Confirm limits: either supply the interval or let the calculator solve intersection points for you.
- Compute to see the integrand, evaluated integral, and both exact and numerical volume results. Copy steps to your notes.
Examples
- Disk about x‑axis: Region under on rotated about x‑axis ⇒ , .
- Washers about x‑axis: Between and on ⇒ , .
- Shells about y‑axis: Region under on rotated about y‑axis ⇒ .
- Shifted axis y=2 (washers): Region between and , , around ⇒ , .
Formula Snippets Ready for Rendering
FAQs
When should I use disks or washers instead of shells?
Use disks/washers when slices perpendicular to the axis yield simple radii; use shells when parallel slices make the integrand simpler.
How do I pick the correct variable of integration?
Integrate with respect to the variable along which you slice. Perpendicular to x‑axis → integrate in x; perpendicular to y‑axis → in y.
What if the region touches the axis of rotation?
The inner radius is zero; the washer method reduces to the disk method.
How do I handle a shifted axis like y = c or x = c?
Measure radii or shell radii as absolute distances to that line, e.g., R = |F(x) − c| and r = |G(x) − c|.
What if curves cross inside my interval?
Split the integral at intersection points and recompute which curve is outer/inner (washers) or height (shells) in each subinterval.
Can I mix functions of x and y?
Yes—rewrite as y = f(x) or x = g(y) to match your chosen orientation; the calculator can assist with conversion.
Do I need exact antiderivatives?
No. The calculator integrates symbolically when possible, otherwise uses numeric methods with controllable precision.
Will the result include units?
Yes. Input units carry through and are cubed. You can optionally convert to liters or gallons.
What are common mistakes?
Forgetting to square radii, mixing up outer/inner curves, using the wrong axis distance, and not splitting at intersections.
Can I verify with another method?
Yes—use shells to cross‑check a washer result or apply Pappus’ centroid theorem when applicable.
Does this support piecewise or absolute‑value functions?
Yes. The region is broken into consistent pieces and summed; absolute values are handled explicitly in radii definitions.
What about revolving around slanted lines?
Translate/rotate coordinates to align with an axis, compute the volume, then interpret back in the original frame.