Washer Method Calculator
Compute volume of a solid of revolution using washers. Enter outer f(x), inner g(x), and limits to integrate accurately today.
Equation Preview
Provide functions and limits to see the substituted integral.
Helping Notes
Fields mirror standard practice: outer function f(x), inner function g(x), and the two limits a and b for integration.
If a > b, the interval is swapped automatically. Use expressions like sqrt(x), x^2+1, sin(x), or constants.
Results
Volume V
Definite Integral (no π)
Interval
What Is a Washer Method Calculator?
A Washer Method Calculator finds the volume of a solid of revolution formed by rotating a planar region around a line (axis). It applies the washer (or disk-with-hole) approach: at each slice perpendicular to the axis, the cross‑section is a circular washer with outer radius and inner radius . The volume equals the integral of the area difference of these washers across the interval. The tool supports rotation about the coordinate axes and about shifted lines such as or , handles functions expressed as or , and returns exact symbolic forms when possible, along with numeric values. Clear, responsive formulas and step‑by‑step setup help students verify limits, radii, and orientation before integrating.
About the Washer Method Calculator
Choose the variable of integration so that slices are perpendicular to the axis of rotation. For rotation about the x‑axis, integrate with respect to x using radii measured vertically; for rotation about the y‑axis, integrate with respect to y using horizontal distances. For a region bounded above by and below by rotated around the x‑axis, the outer and inner radii are typically and when the axis is . With rotation about , switch to functions of y. The calculator also supports piecewise intervals when curves cross, unit conversions for the result, and Riemann‑sum approximations for discrete data.
Washer about the x‑axis (or y=c):
Washer about the y‑axis (or x=c):
Riemann approximation:
Diameters (if given):
Intersection limits:
How to Use This Washer Method Calculator
- Enter the bounding curves and the rotation axis (e.g., around x‑axis, y‑axis, or a shifted line y=c or x=c).
- Pick the correct variable of integration so slices are perpendicular to the axis. The tool auto‑suggests x vs. y.
- Confirm limits: either the interval endpoints or the intersection points of the bounding curves.
- Compute: the calculator builds , integrates symbolically if feasible, and reports exact and numeric volumes.
Examples
- Disk (no hole), x‑axis: Region under from around x‑axis ⇒ , .
- Washer with hole, x‑axis: Between and on ⇒ , .
- About y‑axis (use y): Region between and on around y‑axis ⇒ , .
- Shifted axis y=2: Rotate region under , , about ⇒ , .
Formula Snippets Ready for Rendering
FAQs
When should I use washers instead of shells?
Use washers when slices perpendicular to the axis are easier to describe; use shells when slices parallel to the axis simplify setup.
What if the region touches the axis?
Then the inner radius is zero and the method reduces to the disk method: .
How do I handle a shifted axis like y = c?
Measure radii as distances to that line: ; square them in the integral.
Do I integrate with respect to x or y?
Choose the variable that makes slices perpendicular to the axis and radii easy to express.
Can radii be negative?
No—radii are distances. Use absolute values when measuring from shifted axes or below/above relationships change.
What if curves cross inside the interval?
Split the integral at crossing points and redefine which curve supplies the outer versus inner radius on each subinterval.
How do I find intersection limits?
Solve (or ) for the bounds; the calculator can compute these automatically.
Can the calculator show steps?
Yes, it displays radii definitions, limits, the integrand , and the evaluated integral in symbolic and numeric forms.
Does this work with piecewise or absolute‑value functions?
Yes—split into intervals where expressions are consistent, then sum the resulting volumes.
What units will the volume use?
Units cube: meters → m³, inches → in³, etc. The calculator optionally converts to liters or gallons.