Name: double integrals on curved boundaries
Uploaded: 2026-06-08T09:38:44.708Z
Duration: 1 min 18 s
Channel: JoVE

Consider an irregularly shaped region, D, of a pond which is enclosed within a rectangular park, Region R.Because the boundaries of the pond are curved and irregular, performing integration directly over Region D can be highly complex.To simplify the integration, a new function, g, is defined over the entire rectangle, matching the original function f inside D, and equal to zero everywhere outside the pond&#x27;s boundaries.In practice, the double integral is evaluated as an iterated integral by utilizing variable bounds of integration.If the region is bounded between constant vertical lines on the horizontal axis, moving from a lower boundary curve to an upper boundary curve, it is classified as a type I region.Alternatively, if the region is bounded between constant horizontal lines on the vertical axis, moving from a left boundary curve to a right boundary curve, it is classified as a type II region.By describing these curved boundaries as variable functions, double integrals over any general region can be seamlessly evaluated.

double integrals over general regions

Double integrals are often used to measure quantities distributed across two-dimensional regions, such as rainfall over a lake, heat across a metal plate, or population density over land. In many practical situations, the region of interest does not have straight boundaries and cannot be described conveniently as a rectangle. Instead, the region may have curved or irregular edges. To evaluate integrals over such domains, the region is embedded inside a larger rectangular region where integration methods are easier to apply.Extending Double Integrals to Irregular RegionsLet D represent an irregular region enclosed within a rectangular region R. Suppose f(x,y) describes a quantity distributed over D, such as rainfall intensity over a lake. A new function F(x,y) is defined on the entire rectangle R by\begin{equation*}F(x,y)=\begin{cases}f(x,y), &amp; (x,y)\in D \\0, &amp; (x,y)\in R \text{ but not in } D\end{cases}\end{equation*}This construction allows integration to be performed over the rectangle R while ensuring that only values inside D contribute to the result.Since F(x,y) = 0 outside the irregular region, the surrounding portions of the rectangle add nothing to the integral.The double integral over the general region D is therefore written as\begin{equation*}\iint_D f(x,y)\, dA = \iint_R F(x,y)\, dA\end{equation*}Type I and Type II RegionsIrregular regions are commonly classified into two forms. A type I region is bounded vertically and can be expressed as\begin{equation*}D=\{(x,y)\mid a\le x\le b,\; g_1(x)\le y\le g_2(x)\}\end{equation*}A type II region is bounded horizontally and is written as\begin{equation*}D=\{(x,y)\mid c\le y\le d,\; h_1(y)\le x\le h_2(y)\}\end{equation*}These descriptions convert integration over curved regions into iterated integrals with variable limits, enabling systematic evaluation of quantities distributed over irregular domains.

Double Integrals Over General Regions

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JoVE Core

Calculus

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Multiple Integrals and Applications

KEY TERMS AND CONCEPTS

double integrals on curved boundaries

Double integrals can measure quantities spread across a two-dimensional region, such as rainfall over a lake, heat on a metal plate, or population density over land. In many real situations, the region does not have straight sides. It may have curved or irregular edges instead. That makes the region harder to describe as a simple rectangle.

To handle an irregular region D, it is placed inside a larger rectangular region R. A function f(x,y) gives the quantity on D, such as rainfall intensity over a lake. A new function F(x,y) is then defined on all of R. It equals f(x,y) inside D and 0 everywhere in R that lies outside D.

This setup lets the integral be taken over the easier rectangular region R. The outside parts of the rectangle contribute nothing because F(x,y)=0 there. The double integral over the general region D is written as the double integral over R of F(x,y) dA.

Irregular regions are often described as Type I or Type II regions. A Type I region is bounded vertically. It is written with x running from a to b, while y runs from g1(x) to g2(x).

A Type II region is bounded horizontally. It is written with y running from c to d, while x runs from h1(y) to h2(y). These forms turn double integrals over curved regions into iterated integrals with variable limits. That makes the calculation of quantities over irregular domains more systematic.

Double Integrals on Curved Boundaries

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Double integrals can measure quantities spread across a two-dimensional region, such as rainfall over a lake, heat on a metal plate, or population density over land. In many real situations, the region does not have straight sides. It may have curved or irregular edges instead. That makes the region harder to describe as a simple rectangle. To handle an irregular region D, it is placed inside a larger rectangular region R. A function f(x,y) gives the quantity on D, such as rainfall intensity...

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Learn how double integrals handle irregular regions by using a larger rectangle, a zero-outside function, and Type I and Type II region bounds.

Double Integrals on Curved Boundaries | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

calculus

Double Integrals on Curved Boundaries

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