Calculator
Graph and Sample Values
Submit the form to draw the curves and integrand samples.
Formula Used
Shell method around the y-axis: V = 2π ∫ radius × height dx. For the line x = k, the radius is |x - k|. The height is f(x) - g(x).
Washer method around the y-axis: V = π ∫ (outer radius² - inner radius²) dy. This mode uses curves written as x-values in terms of y.
Numerical rule: Composite Simpson's rule estimates the integral by fitting parabolic arcs through evenly spaced sample points.
How to Use This Calculator
- Choose shell mode when your curves are easier as functions of x.
- Choose washer mode when your curves are easier as functions of y.
- Enter the correct bounds for the variable used by the method.
- Keep the axis value at 0 for rotation about the y-axis.
- Increase slices when curves change quickly or need more precision.
- Press Calculate Volume, then review the result, graph, and sample table.
- Use the CSV and PDF buttons to save your result.
Example Data Table
| Method | Region setup | Bounds | Axis | Expected volume |
|---|---|---|---|---|
| Shell | f(x)=x^2, g(x)=0 |
0 ≤ x ≤ 2 |
x=0 |
8π ≈ 25.1327 |
| Washer | R(y)=sqrt(y), r(y)=0 |
0 ≤ y ≤ 4 |
x=0 |
8π ≈ 25.1327 |
| Shell | f(x)=4-x^2, g(x)=0 |
0 ≤ x ≤ 2 |
x=0 |
8π ≈ 25.1327 |
Y-Axis Rotation Volume Guide
A solid of revolution appears when a flat region turns around an axis. This calculator focuses on rotation around the y-axis. It supports the shell method and the washer method. Both approaches can describe the same shape. The best choice depends on how the region is written.
Why the Method Matters
The shell method works well when curves are given as y values in terms of x. A thin vertical strip rotates around the y-axis. That strip becomes a cylindrical shell. Its radius is the distance from the strip to the axis. Its height is the gap between the upper and lower curves. The calculator multiplies radius, height, and circumference, then integrates.
The washer method works well when curves are given as x values in terms of y. A thin horizontal strip rotates around the y-axis. That strip becomes a washer or disk. The outer radius creates the outside edge. The inner radius removes a hole. The calculator integrates the difference between squared radii.
Accuracy and Setup
Numerical integration is used for flexible inputs. Simpson's rule gives strong accuracy for smooth functions. More slices usually improve the result. Very steep curves, discontinuities, or wrong bounds can reduce accuracy. Always review the sample points and chart before using the answer in final work.
Reading the Graph
The graph is more than decoration. It reveals bound mistakes quickly. It can show reversed curves, negative gaps, and surprising radius changes. Compare the shaded idea in your notes with the plotted curves. Then adjust the entries before trusting the number. This protects final answers from errors.
Common Mistakes
Many errors come from mixing variables. Shell mode expects x-bounds and functions of x. Washer mode expects y-bounds and functions of y. Another error is using a negative height. Put the upper curve first in shell mode. Put the outer radius first in washer mode.
Practical Use
Use this tool for homework checks, design estimates, and quick calculus exploration. It shows formulas, integrand values, and downloadable results. The chart also helps confirm whether the region looks reasonable. For exact symbolic answers, simplify the integral by hand after checking the numerical result here.
FAQs
1. What does rotating about the y-axis mean?
It means a flat region turns around the vertical y-axis. The sweep creates a three-dimensional solid. The calculator estimates the volume of that solid.
2. When should I use the shell method?
Use the shell method when your region is described with functions of x. It is often easier for vertical slices rotated around the y-axis.
3. When should I use the washer method?
Use the washer method when the curves are written as x in terms of y. It works with horizontal slices and radius differences.
4. Why is the axis value editable?
The default axis is x = 0, which is the y-axis. The editable value also lets you test nearby vertical axis rotations.
5. Does this calculator give exact symbolic answers?
No. It gives a numerical estimate using Simpson's rule. Use it to check setup, compare methods, and support manual symbolic work.
6. Why can the signed volume be negative?
A negative signed value usually means a curve order is reversed. Volume is reported as positive, but the warning should be reviewed.
7. How many slices should I choose?
Start with 200 slices. Increase the count for rapidly changing functions. Very high counts can slow the page without much visible gain.
8. Can I export the result?
Yes. After calculation, use the CSV button for table data. Use the PDF button for a compact report with the main result.