--- title: "Areas Between Two Curves" description: "Find where two curves meet, then integrate the difference of the two functions between those limits to get the area enclosed." canonical: https://lightmysky.com/learn/mathematics/areas-between-two-curves-mt_lDBZmnLK1b source: https://lightmysky.com/learn/mathematics/areas-between-two-curves-mt_lDBZmnLK1b.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`. # Areas Between Two Curves Find where two curves meet, then integrate the difference of the two functions between those limits to get the area enclosed. Subject: Mathematics · Area: Calculus & Analysis · Ages 17 to 18 Page: https://lightmysky.com/learn/mathematics/areas-between-two-curves-mt_lDBZmnLK1b ## Ready when they can - Find the intersection points of a line and a parabola - Integrate the difference of the two functions between those limits - Say which function goes first in the subtraction and why ## Lesson: The gap between two curves Before you integrate, find where the curves meet, because those crossing points become your limits. Set the two expressions equal and solve. The line y equals x meets the parabola y equals x squared where x equals x squared, which gives x equals 0 or 1, so the points are (0, 0) and (1, 1). Your checkpoint pair works the same way. Setting x squared equal to x plus 2 gives x equals 2 or negative 1, so the crossings sit at those two x values. **Example.** Each thin strip of the gap is a rectangle with height equal to top minus bottom, so the area is the integral of that difference between your crossings. Between 0 and 1 the line y equals x sits above y equals x squared, and integrating x minus x squared gives 1 over 2 minus 1 over 3, which is 1 over 6. A bigger gap works the same way. The line y equals 6x and y equals x squared cross at 0 and 6, and integrating 6x minus x squared there gives 108 minus 72, which is 36. Your assessment pair closes the set. The line y equals x plus 2 stays above y equals x squared from negative 1 to 2, and integrating the difference gives 9 over 2. Always check which curve goes first in the subtraction. Sketch both graphs, or test one x value between the crossings and compare. Between 0 and 1, testing x equals 1 over 2 gives 1 over 2 against 1 over 8, so y equals x goes first and the setup is the integral of x minus x cubed. If the curves cross inside your interval, one formula cannot hold throughout, because top and bottom swap and signed parts cancel. Split at each crossing, set top minus bottom fresh in every piece, and add the separate areas. **Recap.** Crossings give the limits, top minus bottom gives the height, and a test point plus splitting keeps every piece honest. ## Practice 14 questions on this page, each with its working shown. ## Needs first - [Simultaneous Equations with One Quadratic](https://lightmysky.com/learn/mathematics/simultaneous-equations-with-one-quadratic-mt_D89ql2sJvc) - [The Definite Integral and the Area Under a Curve](https://lightmysky.com/learn/mathematics/the-definite-integral-and-the-area-under-a-curve-mt_S2IO2S7kgd) ## Opens up - [Volumes of Revolution: Disks, Washers and Shells](https://lightmysky.com/learn/mathematics/volumes-of-revolution-disks-washers-and-shells-mt_mzZ22MkWD4) - [Double Integrals over General Regions](https://lightmysky.com/learn/mathematics/double-integrals-over-general-regions-mt_UKS9_AhAWw)