--- 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: 2026-11-12 --- > **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 and interpret intersections of a line and a parabola - Evaluate geometric area using definite integrals, including separate signed pieces - Choose top minus bottom for vertical strips or right minus left for horizontal strips - Use supplied interval boundaries when an area is not enclosed solely by the curves ## Lesson: The gap between two curves **Example.** Find intersections by setting the two rules equal. For y = 3x and y = x², x² - 3x = x(x - 3) = 0 gives x = 0 or 3. Substitute into the line to get (0, 0) and (3, 9). These are full points, not just integration limits. **Example.** For y = x + 12 and y = x², equating gives x² - x - 12 = (x - 4)(x + 3) = 0. The roots are 4 and -3. Compare their signed values to identify the smaller crossing x-coordinate. This uses definite integration. On a <= x <= b where top and bottom keep their order, choose F with F'(x) = top - bottom and evaluate F(b) - F(a). If only the two curves enclose the region, their intersection x-coordinates give the outer limits. If vertical boundaries are supplied, use those instead. Check for crossings inside the interval and split where the order changes. **Example.** A vertical strip has height top minus bottom. For y = 2x and y = x², the enclosed limits are 0 and 2. Integrate 2x - x²: F(x) = x² - x³/3. The area is F(2) - F(0) = 4 - 8/3 = 4/3. **Example.** For y = 4x and y = x², the crossings are 0 and 4 and the line is higher between them. An antiderivative of 4x - x² is 2x² - x³/3. Evaluate at both limits: (32 - 64/3) - 0 = 32/3. Choose the upper function by a sketch or a test value between consecutive crossings. On 0 < x < 1, x⁵ < x, so the height between y = x and y = x⁵ is x - x⁵. A test point must lie inside the interval, not at a crossing. When two curves cross inside the interval, the upper and lower functions exchange places. Split there, subtract lower from upper on each piece, and add the positive areas. A fixed subtraction order would instead allow signed parts to cancel. **Example.** For y = x + 6 and y = x², the enclosed limits are -2 and 3. The line is higher. Integrate x + 6 - x² using F(x) = x²/2 + 6x - x³/3. F(3) = 27/2 and F(-2) = -22/3, so the area is 27/2 - (-22/3) = 125/6. A supplied interval can bound an area even when its endpoints are not intersections. For y = 16 and y = x² restricted to 1 <= x <= 3, the limits are 1 and 3. The curves' intersections at -4 and 4 lie outside this restricted region. A line tangent to a parabola has one distinct intersection. Equating y = x² with y = 8x - 16 gives (x - 4)² = 0. This contact does not by itself enclose a positive-area region. A definite integral is signed. For y = -3 and the x-axis on [0, 2], integrating -3 gives -6, while the geometric area is 6. For area, subtract the lower function from the upper rather than accepting a negative answer. Split wherever f - g changes sign. Within each resulting piece, its sign must stay constant before you turn a signed integral into geometric area by taking its magnitude. If two such pieces have signed integrals 2 and -7, their areas are 2 and 7, giving total area 9. If either piece still contains a sign change, subdivide it before adding magnitudes. **Example.** Horizontal strips use right minus left and integrate with respect to y. Between x = y² and x = 9 for 0 <= y <= 3, the strip width is 9 - y². The area setup is the integral of 9 - y² from y = 0 to 3. Meeting is not always crossing. At a tangential contact, top and bottom may stay in the same order on both sides. For y = (x - 2)² and y = 0, the square stays nonnegative. Splitting at 2 is allowed but is not needed to prevent cancellation. **Recap.** Use supplied boundaries or enclosing intersections for the limits. Integrate top minus bottom, splitting at every change of order, then add the nonnegative areas. *Lesson and assessment explanations written for Marble.* ## 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)