Volume of Revolution Calculator

Calculate the volume of a solid formed by revolving a function around the x-axis or y-axis using integration.

Disk Method
Washer Method
Shell Method

Disk Method Formula: V = π ∫[a,b] [f(x)]² dx (around x-axis)

Used when the region touches the axis of revolution (no hole in the middle).

f(x) =
Enter a function of x (use ^ for exponent, sqrt() for square root)
x²
sin(x)
√x
1/x
x³ - 2x

Washer Method Formula: V = π ∫[a,b] ([R(x)]² - [r(x)]²) dx (around x-axis)

Used when there's a hole in the middle (region doesn't touch the axis).

R(x) =
r(x) =

Shell Method Formula: V = 2π ∫[a,b] (radius)(height) dx

Used when revolving around an axis parallel to the axis of integration.

f(x) =

Understanding Volume of Revolution

The volume of revolution is the volume of a three-dimensional object obtained by rotating a two-dimensional region around an axis. This is a fundamental concept in calculus with applications in physics, engineering, and geometry.

Key Methods for Calculating Volume of Revolution:

  • Disk Method: Used when the region touches the axis of revolution
  • Washer Method: Used when there is a hole in the middle of the solid
  • Shell Method: Used when integrating perpendicular to the axis of revolution

Formulas

Method Formula (around x-axis) When to Use
Disk V = π ∫[a,b] [f(x)]² dx Region touches x-axis
Washer V = π ∫[a,b] ([R(x)]² - [r(x)]²) dx Region has a hole (R = outer radius, r = inner radius)
Shell V = 2π ∫[a,b] x f(x) dx Revolving around y-axis

Step-by-Step Calculation Process

1

Identify the Region: Determine the 2D region to be revolved and sketch it if possible.

2

Choose the Method: Select disk, washer, or shell method based on the axis of revolution and whether there's a hole.

3

Set Up the Integral: Write the appropriate integral formula with the correct limits of integration.

4

Evaluate the Integral: Compute the definite integral using integration techniques.

5

Interpret the Result: The result is the volume of the 3D solid formed by revolution.

Common Examples

Example 1: Volume of a cone

Rotate f(x) = (h/r)x from x=0 to x=r around the x-axis:

V = π ∫[0,r] [(h/r)x]² dx = (1/3)πr²h

Example 2: Volume of a sphere

Rotate f(x) = √(r² - x²) from x=-r to x=r around the x-axis:

V = π ∫[-r,r] [√(r² - x²)]² dx = (4/3)πr³

Example 3: Volume of a torus (donut)

Rotate a circle (x-R)² + y² = r² around the y-axis using washer method:

V = 2π²Rr²

Applications

  • Engineering: Calculating volumes of tanks, pipes, and machine parts
  • Physics: Determining moments of inertia and centers of mass
  • Architecture: Designing domes, arches, and other curved structures
  • Manufacturing: Calculating material needed for rotational molding
  • Medicine: Modeling organs and blood vessels

Frequently Asked Questions

The disk method is used when the region touches the axis of revolution (creating solid disks). The washer method is used when there's a hole in the middle (creating washers or rings). The shell method is used when integrating perpendicular to the axis of revolution (creating cylindrical shells).

The choice often depends on which integral is easier to evaluate. Disk/washer methods integrate along the axis of revolution, while shell methods integrate perpendicular to it. Sometimes one method leads to simpler integrals than the other. If the region is described as y=f(x) and you're revolving around the y-axis, the shell method is often easier.

Yes, the calculator supports functions in terms of both x and y. You can select the integration variable in the advanced options. For functions in terms of y (x = f(y)), simply select 'y' as the variable and adjust your limits accordingly.

If the function has discontinuities or vertical asymptotes within the integration interval, the calculator will detect this and split the integral at the discontinuity points. However, for infinite discontinuities, the volume may be infinite or undefined.

The calculator uses adaptive numerical integration with error control. You can adjust the precision in the advanced options. For well-behaved functions, the accuracy is typically within 0.001% of the true value for the medium precision setting.