Build the arc length formula calculus needs: the labeled ds right triangle, the swept surface band, and the assembled integral solved step by step.
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The Arc Length & Surface-of-Revolution Setup Diagram draws the picture behind the arc length formula calculus uses, L = ∫√(1 + (dy/dx)²) dx. Pick a curve — eight cartesian functions or four AP Calculus BC parametric curves — set the bounds, and optionally choose an axis of revolution (the x-axis, the y-axis, y = k or x = h).
The tool renders the zoomed differential triangle with its dx, dy and ds legs labeled, the curve with its mirrored image and the swept band between them, the revolution radius at a representative slice, and the assembled arc-length and surface integrals with their values. Every derivative comes from a pinned closed-form table and both integrals are evaluated with a fixed composite Simpson rule, so the same inputs always give the same answer.
For y = f(x) on [a, b] the arc length formula is L = ∫√(1 + (dy/dx)²) dx. It comes from adding the hypotenuses of infinitely many right triangles with legs dx and dy, so ds = √(dx² + dy²). For a parametric curve (x(t), y(t)) the same idea gives L = ∫√((dx/dt)² + (dy/dt)²) dt.
Revolving a piece of curve of length ds about an axis sweeps a thin band of lateral area 2πr·ds, where r is the distance from that piece to the axis. Adding the bands gives S = 2π∫r ds — the same ds, with one extra radius factor.
Subtract the axis constant. About the x-axis r = |y| and about y = k it is |y − k|; about the y-axis r = |x| and about x = h it is |x − h|. The curve must stay on one side of that line, otherwise the integral has to be split at the crossing.
The differential triangle is built from the tangent line, so dy = f′(x)·dx rather than the secant rise f(x + dx) − f(x). That is what makes ds a differential and lets the sum of the pieces become an integral in the limit.
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