Arc Length & Solid of Revolution
Arc length and x-axis revolution formulas, evaluated numerically with Simpson integration.
Curvilinear Distance Integration 🖖
Arc length calculates the exact 1D distance along a continuously warped 2D path. Derived directly from the infinitesimal Pythagorean theorem, it integrates the exact square root of the derivative squared, translating non-linear curved trajectories into finite linear distance metrics.
Spinning a curve into a solid 🖖
The second mode of this tool acts like a lathe or a potter's wheel: take the curve y=f(x) and spin it around the x-axis to sweep out a 3D shape. To find its volume, the tool slices the solid into thin discs, each a circle of radius f(x), and adds them up. Spin the line y=x from 0 to 3 and you get a perfect cone of volume 9π — no geometry formula needed, just the integral.
The trumpet you can fill but never paint 🖖
Revolve y=1/x from x=1 to infinity around the x-axis and you get Gabriel's Horn. Its volume converges to exactly π, yet its surface area is infinite. This "painter's paradox" means the horn holds a finite π units of paint, but no finite amount could ever coat its inner wall. Evangelista Torricelli found this in the 1640s, before calculus formally existed, and even he thought it was absurd.
Example problems
- parabola arc - Arc length of y=x^2 on [0,2] is longer than the straight chord.
- semicircle arc - Semicircle arc on [-1,1] approaches pi as expected.
- sphere by revolution - Revolving a semicircle around x-axis yields sphere-like 4pi and 4pi/3 values.
- cone by revolution - Revolving y=x on [0,3] produces a cone with analytic volume 9pi.