--- title: "Arc Length and the Area of a Surface of Revolution" description: "Set up the integrals that measure the length of a plane curve and the area swept when it is spun about an axis." canonical: https://lightmysky.com/learn/mathematics/arc-length-and-the-area-of-a-surface-of-revolution-mt_W4uujT9Z1t source: https://lightmysky.com/learn/mathematics/arc-length-and-the-area-of-a-surface-of-revolution-mt_W4uujT9Z1t.md retrieved: 2026-09-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Arc Length and the Area of a Surface of Revolution Set up the integrals that measure the length of a plane curve and the area swept when it is spun about an axis. Subject: Mathematics · Area: Calculus & Analysis · Ages 19 to 20 Page: https://lightmysky.com/learn/mathematics/arc-length-and-the-area-of-a-surface-of-revolution-mt_W4uujT9Z1t ## Ready when they can - Derive the arc length integrand from a limit of straight-line approximations - Compute the arc length of a curve given in Cartesian and in parametric form - Set up and evaluate a surface-of-revolution area integral about each axis ## Lesson: Measuring curves and the surfaces they sweep Arc length starts from Pythagoras. A tiny piece of curve is nearly straight, with length from dx and dy. Factoring out dx gives the famous integrand: the square root of (1 plus (dy/dx) squared). Integrating adds the pieces exactly. So the formula comes from adding short straight segments and passing to a limit. **Example.** Straight lines keep the integral friendly. For y is (3/4)x, the slope is 3/4, so the integrand is the constant 5/4. From 0 to 4 that gives 5, and from 0 to 8 it gives 10. A horizontal line has zero slope, so y is 5 from 2 to 9 has length 9 minus 2, which is 7. Constant integrand means length is just rate times width. Revolving a curve sweeps out bands, and each band unwraps to nearly a rectangle: circumference times width. The circumference is 2 pi times the radius, and the width is the arc length element. About the x-axis the radius is the function value f(x), and about the y-axis it is the distance to that axis, g(y). Only the integral with both the 2 pi f(x) factor and the square root is the surface area. **Tip.** Parametrise curves you cannot write as y is f(x), then integrate speed: the square root of ((dx/dt) squared plus (dy/dt) squared). Sanity check answers by revolving a straight segment, which gives a cone or cylinder with geometry you already know. The same idea covers wider ground with no new machinery. **Recap.** Build length from tiny straight pieces, spin bands for area, and parametrise when y will not cooperate. ## Practice 14 questions on this page, each with its working shown. ## Needs first - [Integration by Substitution](https://lightmysky.com/learn/mathematics/integration-by-substitution-mt_jH-j_XsdC7) - [Volumes of Revolution: Disks, Washers and Shells](https://lightmysky.com/learn/mathematics/volumes-of-revolution-disks-washers-and-shells-mt_mzZ22MkWD4) ## Opens up - [Change of Variables and the Jacobian](https://lightmysky.com/learn/mathematics/change-of-variables-and-the-jacobian-mt_1_LRE2cJft)