Name: evaluating triple integrals by region
Uploaded: 2026-06-08T09:38:52.664Z
Duration: 1 min 28 s
Channel: JoVE

Triple integrals over general regions describe accumulation across a bounded solid in three-dimensional space. The region E may have curved or irregular boundaries, so a systematic way is needed to define the bounds.To define the integral over E, the solid is placed inside a larger rectangular box. A new function matches the original function inside E and is zero outside the solid. This defines the triple integral over E using the surrounding box. But for practical evaluation, the solid can be described using vertical slices.A practical example is estimating the amount of ore in an underground deposit whose top and bottom surfaces are given by two functions.The projection of this deposit onto the horizontal plane forms a base region, D. Each point in D gives a vertical segment through the solid, from the lower surface to the upper surface.The triple integral is then evaluated as an iterated integral using vertical slices. In the inner integral, values are added along the vertical segments. These totals are then added over the base region D.So, the triple integral gives the total amount of ore within the solid.

triple integrals over general regions

Triple integrals over general bounded regions extend the concept of double integrals from planar domains to three-dimensional solids. A solid region E in space is commonly enclosed within a rectangular box B, and a continuous function f(x, y, z) is integrated over the region by defining F such that it coincides with f on E and is zero outside the solid. The triple integral is therefore expressed as\begin{equation*}\iiint_E f(x,y,z) dV \end{equation*}The existence of the integral requires that f be continuous and that the boundary of E be sufficiently smooth. Similar to double integrals, the evaluation of triple integrals depends strongly on the geometric description of the region.A common class of solids is the type I region, where the variable z is bounded between two continuous surfaces:\begin{equation*}E=\{(x,y,z)\mid (x,y)\in D; u_1(x,y)\le z\le u_2(x,y)\}\end{equation*}Here, D represents the projection of the solid onto the xy-plane. The lower boundary of the solid is given by z = u1(x, y), while the upper boundary is described by z = u2(x, y). In this case, the triple integral can be rewritten as an iterated integral:\begin{equation*}\iiint_E f(x,y,z)\, dV=\iint_D\left[\int_{u_1(x,y)}^{u_2(x,y)}f(x,y,z)\, dz\right] dA\end{equation*}If the projection D is a type I plane region,\begin{equation*}a\le x\le b,\qquad g_1(x)\le y\le g_2(x)\end{equation*}The integral becomes\begin{equation*}\int_a^b\int_{g_1(x)}^{g_2(x)}\int_{u_1(x,y)}^{u_2(x,y)}f(x,y,z) dz\, dy\, dx\end{equation*}Alternatively, when D is a type II plane region described by\begin{equation*}c\le y\le d,\qquad h_1(y)\le x\le h_2(y)\end{equation*}The order of integration changes accordingly:\begin{equation*}\int_c^d\int_{h_1(y)}^{h_2(y)}\int_{u_1(x,y)}^{u_2(x,y)}f(x,y,z) dz\, dx\, dy\end{equation*}These representations provide flexible methods for evaluating integrals over complex three-dimensional solids.

Triple Integrals over General Regions

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JoVE Core

Calculus

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Multiple Integrals and Applications

KEY TERMS AND CONCEPTS

evaluating triple integrals by region

Triple integrals over bounded solid regions are used to measure a continuous function across a three-dimensional shape. The solid region E is often placed inside a rectangular box B. To handle the region cleanly, the function f(x,y,z) is extended to a new function F that matches f inside E and is zero outside the solid. The triple integral is written as ∭_E f(x,y,z) dV.

For the integral to exist, the function f must be continuous. The boundary of E also needs to be smooth enough. As with double integrals, the way the integral is evaluated depends on how the solid is described geometrically. That description is the key to choosing the order of integration.

A common case is a type I region. In this case, z lies between two continuous surfaces. The solid is written as E = {(x,y,z) | (x,y) in D; u1(x,y) ≤ z ≤ u2(x,y)}. Here, D is the projection of the solid onto the xy-plane. The lower surface is z = u1(x,y), and the upper surface is z = u2(x,y).

For a type I region, the triple integral can be rewritten as an iterated integral. The inner integral is taken with respect to z from u1(x,y) to u2(x,y), and then the result is integrated over D. If D is a type I plane region, it can be described by a ≤ x ≤ b and g1(x) ≤ y ≤ g2(x). Then the integral becomes ∫_a^b ∫_{g1(x)}^{g2(x)} ∫_{u1(x,y)}^{u2(x,y)} f(x,y,z) dz dy dx.

The same region can also be handled when the projection D is a type II plane region. In that case, D is written using c ≤ y ≤ d and h1(y) ≤ x ≤ h2(y). The order of integration changes to match that description. The iterated integral becomes ∫_c^d ∫_{h1(y)}^{h2(y)} ∫_{u1(x,y)}^{u2(x,y)} f(x,y,z) dz dx dy. These forms give flexible ways to evaluate integrals over complex three-dimensional solids.

Evaluating Triple Integrals by Region

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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

Evaluating Triple Integrals by Region

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