Volume of Revolution (Disk Method) Calculator

How it is calculated

The line y = m·x + c is spun around the x-axis between x = a and x = b. At each x the cross-section is a disk of radius |y|, so the volume is the integral of its area:

V = π ∫ from a to b of (m·x + c)² dx

The tool uses the expanded antiderivative F(x) = m²x³ ÷ 3 + m·c·x² + c²·x, then integral = F(b) − F(a) and V = π × integral, shown to four decimals. It also reports the interval length b − a. The interval must have b greater than a; otherwise no result is shown. Because the radius is squared, a line that dips below the axis still gives a positive volume, and the units are whatever length unit your numbers share, cubed. Setting m = 0 gives a cylinder of radius |c|.

Common mistakes

• Entering a vertical line or a curve. Only a straight line y = m·x + c is supported, so something like y = x² has to be handled by a different calculator.
• Swapping the limits. If b is not greater than a, the tool returns no result instead of a negative volume.
• Expecting the area under the line instead of the volume. The integrand here is the radius squared, which is why π appears and the answer is cubic.
• Assuming a line that crosses the x-axis gives a volume of zero or a difference of two parts. The radius is squared, so both sides add up as positive volume (y = x − 2 from 0 to 4 gives 16.7552, two equal cones).
• Mixing units, such as a slope in cm per m. Keep x and y in the same unit; the result is that unit cubed.