---
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: 2026-09-12
---

> **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`.

# 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 a volume under a surface over a bounded region

## Lesson: Integrating between two curves

A general region is trapped between two curves, so the inner limits become functions. If y runs from a lower curve to an upper curve while x runs from a to b, write the iterated integral with y inside. Between y = x squared and y = x on [0, 1], the line is on top, so y runs from x squared to x.

**Example.** Reversing means describing the same region with the other variable outside. The triangle with x outside 0 to 1 and y from x to 1 is really 0 <= x <= y <= 1. With y outside 0 to 1, x runs from 0 to y. Both sets of limits change, so redraw the region.

Evaluate inside out, substituting limits before moving outward. Integrating 1 over a region returns its area: the 2 by 3 rectangle gives 6, and the integral from x = 0 to 2 of y = 0 to x of 1 gives 2. Volume works the same way: height z = 2 over the unit square gives 2.

**Tip.** Picking the wrong order makes life hard. When horizontal slices cut the region into pieces with different tops, try dxdy instead. Check any setup with a constant: integrating 1 must return the geometric area of the region.

**Recap.** Sketch the region, put functional limits inside with constants outside, and reverse the order whenever the shape fights back.

## 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)
- [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)