Find rotated solid volume using reliable shell and washer inputs. Enter functions, bounds, and integration settings. Review results with export tools easily.
| Case | Method | Function or Radius Input | Bounds | Approximate Volume |
|---|---|---|---|---|
| 1 | Shell | y = x^2 | x: 0 to 2 | 25.132741 |
| 2 | Shell | y = x + 1 | x: 1 to 3 | 67.020643 |
| 3 | Washer | R(y)=3, r(y)=1 | y: 0 to 4 | 100.530965 |
| 4 | Washer | R(y)=2+y, r(y)=y | y: 0 to 2 | 50.265482 |
The shell method is useful when the function is written as y = f(x). Rotate the region around the y-axis. Use the radius x and the shell height f(x). Then calculate V = 2π ∫ x f(x) dx over the chosen x-bounds.
The washer method is useful when the radius is easier to express as x = R(y) and x = r(y). Rotate around the y-axis. Then use V = π ∫ (R(y)² − r(y)²) dy over the selected y-bounds.
This calculator evaluates the integral numerically with Simpson’s Rule. Higher interval counts usually improve smooth results for well-behaved functions.
Select the method that matches your equation format. Use shell when you know y as a function of x. Use washer when you know outer and inner x-values as functions of y.
Enter the bounds carefully. Add the unit label if needed. Set decimal places and integration intervals. Click the calculate button. The volume result appears above the form. Then export the result as CSV or PDF.
This calculator finds the volume of a solid of revolution. The solid is formed by rotating a region around the y-axis. It supports two common methods. These are the shell method and the washer method. Both are widely used in calculus and applied mathematics.
Some functions are easier to write as y = f(x). In that case, the shell method is often faster. It uses cylindrical shells. Each shell has a radius and a height. The radius is the x-value. The height comes from the function value. This creates an efficient setup for many problems.
Other problems are easier when x is written in terms of y. Then the washer method works better. It uses outer and inner radii. The area of each cross section is found first. Then the calculator integrates those areas across the chosen interval. This gives the total volume.
Not every function has an easy antiderivative. Numeric integration solves that issue. This calculator uses Simpson’s Rule. It samples many points across the interval. Then it combines them into an accurate estimate. Increasing the interval count can improve precision. This is useful for curved or mixed expressions.
Students can use it for homework checks and revision. Teachers can use it for demonstrations. Engineers and analysts can use it for fast geometry estimates. It also helps anyone comparing two setup methods for the same rotation problem.
The result area shows the method, formula, integration model, and final volume. The page also includes an example table, formula notes, and exports. These additions make the tool practical for both learning and reporting tasks.
Use the shell method when your region is easier to describe with y = f(x). It is especially useful for rotation around the y-axis.
Use the washer method when the outer and inner radii are easier to write as x-values depending on y.
No. It gives a numerical approximation. That is useful when the integral is difficult or impossible to solve by hand.
You can enter common expressions such as x^2, sin(x), cos(x), sqrt(x), log(x), and mixed algebraic forms.
The interval count controls numeric precision. Higher values usually improve the estimate, though they may slightly increase calculation time.
Yes. The calculator accepts negative or positive bounds. Make sure the selected interval matches the actual region you want to rotate.
The result uses the unit label you enter. Because volume is three-dimensional, the displayed unit is shown in cubic form.
Yes. You can export the visible result as a CSV file or create a PDF version for sharing, printing, or record keeping.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.