Volume of Revolution Calculator

This calculator computes the volume — and surface area — of a solid formed by revolving a linear or quadratic curve, using either the Disk Method (V = π∫f(x)²dx, spinning around the x-axis) or the Shell Method (V = 2π∫x·f(x)dx, spinning around the y-axis), both evaluated numerically with Simpson's rule across 200 subintervals. Among the numeric inputs, x End has the largest measured effect on Volume at the calculator's default linear curve and Disk Method, ahead of Coefficient a — because Volume scales with roughly the cube of the integration bound (the disk method squares f(x) and then integrates over x) versus roughly the square for a proportional change in Coefficient a. x Start is also left out of the numeric ranking, because it defaults to 0 (a percentage nudge of zero stays zero), and Function Type and Method are both selectors rather than plain numbers. Neither selector dominates Volume outright, though, and which one comes closer depends on the other: under the default Disk Method, switching Function Type from linear to quadratic swings Volume by more than any numeric input does, because squaring a quadratic grows much faster than squaring a linear function over the same bound — but under the Shell Method, which only multiplies f(x) by x instead of squaring it, that same switch moves Volume by noticeably less, and nudging x End actually produces the bigger swing there.

So the single biggest lever on Volume is Function Type at the Disk Method default, but x End at the Shell Method. One thing the sensitivity check cannot show: the final volume is always forced non-negative with Math.abs(volume) before rounding, regardless of which method or bounds produced it — for the Shell Method in particular, where the raw integral 2π·x·f(x) can legitimately go negative across part of the interval, that absolute value can mask a genuine partial cancellation between regions rather than reporting the signed result.