--- title: "Double Integrals over General Regions" description: "Integrate over a plane region by iterating two single integrals, with the inner limits describing the region. Choosing the order of integration is often the whole problem." canonical: https://lightmysky.com/learn/mathematics/double-integrals-over-general-regions-mt_UKS9_AhAWw source: https://lightmysky.com/learn/mathematics/double-integrals-over-general-regions-mt_UKS9_AhAWw.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`. # Double Integrals over General Regions Integrate over a plane region by iterating two single integrals, with the inner limits describing the region. Choosing the order of integration is often the whole problem. Subject: Mathematics · Area: Calculus & Analysis · Ages 19 to 20 Page: https://lightmysky.com/learn/mathematics/double-integrals-over-general-regions-mt_UKS9_AhAWw ## Ready when they can - Set up an iterated integral for a region bounded by two curves - Reverse the order of integration and redraw the region to find the new limits - Compute volume from a nonnegative height function and distinguish it from signed integrals of sign-changing functions ## Lesson: Integrating between two curves For a region with one vertical span at each x, write lower y <= y <= upper y and put dy inside. The outer x-range is constant. Between y = x² and y = 3x on 0 <= x <= 3, the inner limits are x² and 3x. Regions with several spans may need separate integrals. **Example.** Integrating 1 gives area. For -1 <= x <= 1 and 0 <= y <= 4, integrate 1 with respect to y to get 4. Then integrate 4 from x = -1 to 1, giving 4(1 - (-1)) = 8. For a nonnegative height z = f(x, y), the double integral over the base gives volume above the xy-plane. A constant height multiplies the base area. Height 3 over a base of area 2 gives volume 6. A negative or sign-changing function gives a signed result instead. **Example.** Reverse order by describing the same region with horizontal spans. The triangle 0 <= x <= 2, x <= y <= 2 becomes 0 <= y <= 2, 0 <= x <= y. Between y = x² and y = 3x, horizontal spans are y/3 <= x <= √y with 0 <= y <= 9. **Example.** With functional inner limits, evaluate them before moving outward. For the integral of 1 with 0 <= x <= 3 and 0 <= y <= 2x, the inner integral gives 2x. The outer integral gives [x²] from 0 to 3 = 9. **Example.** Hold x fixed while integrating with respect to y. On 0 <= x <= 2, 0 <= y <= 1, the inner integral of x + 2y is [xy + y²] from 0 to 1 = x + 1. Integrating x + 1 from 0 to 2 gives [x²/2 + x] = 4. dy dx uses vertical slices with lower and upper y-boundaries. If those boundaries change formula, split the x-range. Alternatively, try dx dy, which uses horizontal slices with left and right x-boundaries. Rebuild the limits; changing order does not guarantee that one span will suffice. Ordinary area uses lower-to-upper inner limits. Reversing those limits changes the sign of the integral. For a nonnegative integrand, a negative result is a reason to check whether a span was written backward. A varying height must remain inside the integral. Over 0 <= x <= 2 and 0 <= y <= x, height z = x gives inner integral x × x = x². The outer integral is [x³/3] from 0 to 2 = 8/3. Positive and negative heights can cancel in a signed double integral. To find geometric volume between a sign-changing surface and the xy-plane, split where its sign changes and integrate its magnitude instead. To check whether a point belongs to a region, substitute its coordinates into every boundary inequality. It belongs only when all inequalities hold. For 0 <= y <= 3 and y <= x <= 5, check the y-range and then both x-boundaries. **Recap.** Describe each span, match it to dy dx or dx dy, and integrate inside out. Split where boundaries change. A nonnegative height gives volume; a sign-changing height gives a signed result. *Lesson and assessment explanations written for Marble.* ## Practice 14 questions on this page, each with its working shown. ## Needs first - [Areas Between Two Curves](https://lightmysky.com/learn/mathematics/areas-between-two-curves-mt_lDBZmnLK1b) ## Optional connections - [Riemann Sums and the Definite Integral as a Limit](https://lightmysky.com/learn/mathematics/riemann-sums-and-the-definite-integral-as-a-limit-mt_pwa2SRU6P9) ## Opens up - [Product Measure and Fubini's Theorem](https://lightmysky.com/learn/mathematics/product-measure-and-fubinis-theorem-mt_1VzzVmMjTB) - [Double Integrals in Polar Coordinates](https://lightmysky.com/learn/mathematics/double-integrals-in-polar-coordinates-mt_93e-KSzQMT) - [Green's Theorem in the Plane](https://lightmysky.com/learn/mathematics/greens-theorem-in-the-plane-mt_UDgivmPVEJ) - [Joint Distributions, Covariance and Independence](https://lightmysky.com/learn/mathematics/joint-distributions-covariance-and-independence-mt_YyTFTAqhSF)