13.1 Definition and properties of triple integrals

Triple integrals expand double integrals to three dimensions, allowing integration over solid regions. They're written as $\iiint_D f(x, y, z) \, [dV](https://www.fiveableKeyTerm:dv)$, where $f(x, y, z)$ is the integrand and $D$ is the domain. This concept is crucial for understanding volume and mass in 3D space.

Evaluating triple integrals often involves iterated integrals, breaking them into nested single integrals. Fubini's theorem states that for continuous functions, the order of integration doesn't matter. This flexibility helps choose the most convenient approach for solving complex problems.

Definition and Notation

Triple Integrals and Notation

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• Triple integrals extend the concept of double integrals to three dimensions, allowing for integration over a three-dimensional solid region
• The triple integral of a function $f(x, y, z)$ over a domain $D$ is denoted as $\iiint_D f(x, y, z) \, dV$
• $f(x, y, z)$ is called the integrand, which is the function being integrated over the three-dimensional domain
• $dV$ represents the volume element in the integral, indicating integration with respect to volume

Domain of Integration

• The domain of integration, denoted as $D$, is the three-dimensional solid region over which the triple integral is evaluated
• The domain can be described using inequalities involving $x$, $y$, and $z$ coordinates
• Common shapes for the domain include rectangular boxes, spheres, cylinders, and more complex solids bounded by surfaces
• For example, a rectangular box domain can be defined as $a \leq x \leq b$, $c \leq y \leq d$, and $e \leq z \leq f$

Cartesian Coordinates

• Triple integrals are often evaluated using Cartesian coordinates $(x, y, z)$, where $x$, $y$, and $z$ represent the position of a point in three-dimensional space
• The Cartesian coordinate system consists of three mutually perpendicular axes: the $x$-axis, $y$-axis, and $z$-axis
• The origin $(0, 0, 0)$ is the point where all three axes intersect
• Cartesian coordinates allow for the description of the domain and the evaluation of the triple integral

Iterated Integrals and Fubini's Theorem

Iterated Integrals

• An iterated integral is a way to evaluate a triple integral by breaking it down into a sequence of single integrals
• The triple integral $\iiint_D f(x, y, z) \, dV$ can be written as an iterated integral in the form $\int_a^b \int_c^d \int_e^f f(x, y, z) \, dz \, dy \, dx$
• The innermost integral is evaluated first, followed by the middle integral, and finally the outermost integral
• The bounds of each integral depend on the domain of integration and the order of integration chosen

Order of Integration

• The order of integration refers to the sequence in which the single integrals are evaluated in an iterated integral
• The order of integration can be $dz \, dy \, dx$, $dz \, dx \, dy$, $dy \, dz \, dx$, $dy \, dx \, dz$, $dx \, dy \, dz$, or $dx \, dz \, dy$
• The choice of the order of integration depends on the domain and the integrand, and different orders may lead to different forms of the integral
• For example, $\int_0^1 \int_0^{1-x} \int_0^{1-x-y} f(x, y, z) \, dz \, dy \, dx$ and $\int_0^1 \int_0^{1-z} \int_0^{1-y-z} f(x, y, z) \, dx \, dy \, dz$ represent the same triple integral but with different orders of integration

Fubini's Theorem

• Fubini's theorem states that if $f(x, y, z)$ is continuous over the domain $D$, then the value of the triple integral $\iiint_D f(x, y, z) \, dV$ is independent of the order of integration
• This means that if the conditions of Fubini's theorem are satisfied, the iterated integrals with different orders of integration will yield the same result
• Fubini's theorem allows for flexibility in choosing the most convenient order of integration based on the domain and integrand
• For example, if $f(x, y, z)$ is continuous over a rectangular box domain, the triple integral can be evaluated using any of the six possible orders of integration, and the result will be the same

Properties of Triple Integrals

Linearity Property

• The linearity property states that for constants $a$ and $b$ and integrable functions $f(x, y, z)$ and $g(x, y, z)$ over the domain $D$:
  • $\iiint_D [af(x, y, z) + bg(x, y, z)] \, dV = a \iiint_D f(x, y, z) \, dV + b \iiint_D g(x, y, z) \, dV$
• This property allows for the integration of a sum or difference of functions by integrating each function separately and then combining the results
• The linearity property also applies to constant multiples of functions, where the constant can be factored out of the integral
• For example, if $f(x, y, z) = x^2 + y^2 + z^2$ and $g(x, y, z) = xyz$, then $\iiint_D [3f(x, y, z) - 2g(x, y, z)] \, dV = 3 \iiint_D f(x, y, z) \, dV - 2 \iiint_D g(x, y, z) \, dV$

Additivity Property

• The additivity property states that if $D_1$ and $D_2$ are two non-overlapping domains, and $D = D_1 \cup D_2$, then for an integrable function $f(x, y, z)$:
  • $\iiint_D f(x, y, z) \, dV = \iiint_{D_1} f(x, y, z) \, dV + \iiint_{D_2} f(x, y, z) \, dV$
• This property allows for the evaluation of a triple integral over a domain by splitting it into two or more non-overlapping subdomains, evaluating the integral over each subdomain, and then adding the results
• The additivity property is useful when the domain of integration can be decomposed into simpler subdomains, making the evaluation of the triple integral more manageable
• For example, if a domain $D$ is the union of a sphere and a cube that do not overlap, the triple integral over $D$ can be found by evaluating the integrals over the sphere and the cube separately and then adding the results