Name: double integrals as cross-sections
Uploaded: 2026-06-08T09:38:57.454Z
Duration: 1 min 28 s
Channel: JoVE

A double integral finds the total accumulation of a function over a region. It is evaluated using an iterated integral, which expresses the double integral as two single integrals.For a function over a rectangle in the xy-plane, holding x constant means slicing the region perpendicular to the x-axis and integrating with respect to y. This result is now a function of x, and integrating it with respect to x gives the total accumulation. This process also works in reverse: first along x, then y.Fubini&#x2019;s Theorem governs this process. It states that for a continuous function on a rectangle, both orders of integration give the same result.This concept applies to real-world problems, such as finding the mass of a thin sheet with variable density.In this case, integrating the density with respect to y gives the mass of a thin strip. Integrating these strip masses with respect to x gives the total mass.Similarly, integrating with respect to x first and then y gives the same value. Fubini&#x2019;s Theorem ensures that either order gives the same result, so the order of integration that makes the calculation easier can be chosen.

iterated integrals and fubini's theorem

A double integral generalizes the concept of a single-variable integral to functions of two variables, enabling the computation of the volume beneath a surface z = f(x, y) over a planar region R . For a rectangular region defined by a &#x2264; x &#x2264; b and c &#x2264; y &#x2264; d, and for functions continuous on this domain, the double integral can be evaluated as an iterated integral. This approach simplifies computation by reducing the problem to successive integrations with respect to one variable at a time.Iterated Integration and Cross-sectional AreaHolding one variable constant during integration yields a cross-sectional area function. For instance, fixing x and integrating f(x, y) with respect to y over the interval [c, d] defines the function\begin{equation*}A(x) = \int_{c}^{d} f(x,y)\, dy\end{equation*}which represents the area of a vertical slice through the solid at position x. The total volume under the surface is then obtained by integrating this area along the x-axis:\begin{equation*}\iint_{R} f(x,y)\, dA = \int_{a}^{b} A(x)\, dx = \int_{a}^{b} \int_{c}^{d} f(x,y)\, dy\, dx\end{equation*}A similar argument applies when integrating first with respect to x and then with respect to y, using horizontal cross-sections.Fubini&#x2019;s TheoremFubini&#x2019;s Theorem formalizes the validity of iterated integration. If f is continuous on a rectangular region R, then the double integral over R can be computed as:\begin{equation*}\iint_{R} f(x,y)\, dA = \int_{a}^{b} \int_{c}^{d} f(x,y)\, dy\, dx = \int_{c}^{d} \int_{a}^{b} f(x,y)\, dx\, dy\end{equation*}This result ensures that the order of integration does not affect the value of the integral, providing flexibility in evaluating volumes and other quantities such as mass when density varies over a planar region.

Iterated Integrals and Fubini's Theorem

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Calculus

حساب التفاضل والتكامل

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Multiple Integrals and Applications

التكاملات المتعددة وتطبيقاتها

KEY TERMS AND CONCEPTS

المصطلحات والمفاهيم الرئيسية

التكاملات المتكررة ونظرية فوبيني

double integrals as cross-sections

Double integrals help find the volume under a surface z = f(x,y) over a region in the plane. They extend the idea of a single-variable integral to functions of two variables. For a rectangular region with a ≤ x ≤ b and c ≤ y ≤ d, and for functions that are continuous on that region, the double integral can be written as an iterated integral. This breaks one harder problem into two simpler steps.

A useful way to understand this method is through cross-sectional area. If x is held constant, then integrating f(x,y) with respect to y from c to d gives a slice area function A(x) = ∫c^d f(x,y) dy. This area represents a vertical slice through the solid at position x. The total volume is then found by integrating these slice areas along the x-axis: ∬R f(x,y) dA = ∫a^b A(x) dx = ∫a^b ∫c^d f(x,y) dy dx.

The same idea also works if the order is reversed. You can hold y constant and use horizontal cross-sections, then integrate with respect to x first and y second. This gives another iterated form for the same double integral. The setup depends on whether vertical or horizontal slices are easier to compute.

Fubini’s Theorem explains why this works for continuous functions on a rectangular region. It states that the double integral over R can be computed as either ∫a^b ∫c^d f(x,y) dy dx or ∫c^d ∫a^b f(x,y) dx dy. The order of integration does not change the value. That flexibility is useful when finding volume and other quantities, such as mass when density changes across a planar region.

Double Integrals as Cross-Sections

introduction to double integrals

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Double integrals help find the volume under a surface and over a region in the xy-plane. They extend single-variable integration, which finds area under a curve on an interval, to functions of two variables. This makes them useful for models in mathematics, physics, and engineering that need three-dimensional thinking. Single-variable integration uses thin rectangles to approximate area under y = f(x) from a to b. The definite integral is written as a limit of a sum of these rectangle areas.

the midpoint rule for double integrals

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Double integrals help find the volume under a surface z = f(x,y) over a region in the plane. They extend the idea of a single-variable integral to functions of two variables. For a rectangular region with a ≤ x ≤ b and c ≤ y ≤ d, and for functions that are continuous on that region, the double integral can be written as an iterated integral. This breaks one harder problem into two simpler steps. A useful way to understand this method is through cross-sectional area. If x is held constant, then...

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double integrals over general regions

Double Integrals on Curved Boundaries

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changing the order of integration in double integrals

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triple integrals in rectangular coordinates

triple-integrals-in-rectangular-coordinates

Triple Integrals for Volume and Mass

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triple integrals over general regions

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changing the order of integration in triple integrals

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triple integrals in cylindrical coordinates

triple-integrals-in-cylindrical-coordinates

Volume of Cylinders with Triple Integrals

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triple integrals in spherical coordinates

triple-integrals-in-spherical-coordinates

Finding Sphere Volume with Spherical Wedges

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change-of-variables-in-multiple-integrals

Using the Jacobian to Simplify Integrals

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moments and centers of mass

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real-life applications of multiple integrals

real-life-applications-of-multiple-integrals

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key terms and concepts

Learn how double integrals use cross-sectional area and iterated integrals to find volume under a surface, with Fubini’s Theorem for switching order.

Double Integrals as Cross-Sections | 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 as Cross-Sections

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