Solve solid volume integrals fast. Use washer, shell, and area methods. Export clean results and learn each step clearly.
This sample uses the washer method with x, outer function x^2, inner function 0, and limits from 0 to 2.
| Sample Point | Variable Value | Integrand Value |
|---|---|---|
| 1 | 0.000000 | 0.000000 |
| 2 | 0.400000 | 0.080425 |
| 3 | 0.800000 | 1.286796 |
| 4 | 1.200000 | 6.514805 |
| 5 | 1.600000 | 20.589273 |
| 6 | 2.000000 | 50.265482 |
The calculator estimates volume with definite integration and Simpson’s Rule. It supports cross-sectional area, washer, and shell setups.
Use V = ∫ A(x) dx or V = ∫ A(y) dy when the area of each slice is known.
Use V = π ∫ [R² - r²] d(variable). Here, R is the outer radius and r is the inner radius.
Use V = 2π ∫ radius × height d(variable). The radius is the distance from the shell to the axis.
Simpson’s Rule divides the interval into many even subintervals. It then combines weighted sample values for an accurate estimate.
Select a volume method first. Enter the lower and upper limits. Add the main function. Add the inner function only when your setup needs a hole or inner boundary.
For shell problems, enter the shell height as the main function. Enter the axis constant when the solid rotates around lines like x = 2 or y = 1.
Choose enough Simpson intervals for stable output. Press calculate to show the result below the header. Use the CSV button for data export. Use the PDF button to save the page as a PDF file.
Integral volume helps measure three-dimensional solids. It works well for curved shapes. Standard formulas do not always fit those shapes. Integration solves that gap. It turns many thin slices into one reliable total.
The washer method is useful for hollow solids. The shell method fits rotating regions well. The cross-sectional area method is ideal when each slice area is known. These methods connect geometry with calculus in a clear way.
Use washers when a solid forms by rotation and creates circular slices with holes. You need an outer radius and an inner radius. Squaring both radii gives the cross-sectional ring area. Integration adds all rings together.
Use shells when thin cylindrical layers are easier to describe. This method often avoids solving for inverse functions. The radius measures distance from the axis. The height measures the region span. Their product drives the volume integrand.
Some functions are hard to integrate by hand. Simpson’s Rule gives a strong numerical estimate. It samples points across the interval and applies weighted averaging. This makes the calculator practical for advanced study and quick verification.
A good calculator supports formulas, examples, and exports. It helps students test homework steps. It also helps teachers check setup quality. Engineers and analysts can estimate geometric quantities faster with consistent numeric output and reusable tables.
It computes approximate solid volume from definite integrals. You can use washer, shell, or cross-sectional area methods. The result is numerical.
You can enter expressions using x or y. Supported items include pi, sin, cos, tan, sqrt, abs, exp, log, and powers.
You need it for washer problems with a hollow center. If there is no inner boundary, enter 0.
They are even subintervals used by Simpson’s Rule. More intervals often improve accuracy, especially for curved functions.
Yes. Enter the axis constant. This supports lines such as x = 2 or y = 3 in shell setups.
Negative values usually come from reversed limits or a negative integrand. Swap the limits or review the entered expressions.
Yes. Use the CSV button for spreadsheet-friendly output. Use the PDF button to save the printed page as a PDF.
Yes. It helps you compare methods, test boundaries, inspect integrands, and understand how slicing builds solid volume.
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.