5.4 Triple Integrals

3 min read•june 24, 2024

Triple integrals extend double integrals to three dimensions, allowing us to calculate volumes and other properties of 3D regions. We'll learn how to set up and compute these integrals over rectangular and bounded regions, optimizing the for efficiency.

We'll also explore applications like finding average values of functions over 3D spaces. This powerful tool connects to earlier concepts of and , opening doors to advanced calculations in physics and engineering.

Triple Integrals

Integrable functions over rectangular regions

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  • Continuous functions are integrable over a closed, bounded rectangular region RR in three-dimensional space
  • Piecewise continuous functions with a finite number of discontinuities are integrable over a closed, bounded rectangular region RR
  • Bounded functions satisfying m≤f(x,y,z)≤Mm \leq f(x, y, z) \leq M for all (x,y,z)(x, y, z) in RR are integrable over a closed, bounded rectangular region RR

Computation of triple integrals

  • The ∭Rf(x,y,z)[dV](https://www.fiveableKeyTerm:dV)\iiint_R f(x, y, z) [dV](https://www.fiveableKeyTerm:dV) represents the volume under the graph of f(x,y,z)f(x, y, z) over the region RR
  • Compute triple integrals using iterated integration by integrating with respect to one variable at a time
    • Choose the order of integration based on the region RR and the function f(x,y,z)f(x, y, z)
    • ∭Rf(x,y,z)dV=∫ab∫cd∫eff(x,y,z)dzdydx\iiint_R f(x, y, z) dV = \int_a^b \int_c^d \int_e^f f(x, y, z) dz dy dx
      1. Integrate with respect to zz first, treating xx and yy as constants
      2. Integrate with respect to yy, treating xx as a constant
      3. Integrate with respect to xx
  • Triple integrals are an example of multiple integration, which involves integrating over multiple variables

Triple integrals over bounded regions

  • Express the region RR using a set of inequalities in terms of xx, yy, and zz (0≤x≤1,0≤y≤x,0≤z≤y0 \leq x \leq 1, 0 \leq y \leq x, 0 \leq z \leq y)
  • Determine the limits of integration for each variable based on the inequalities
    • 0≤x≤10 \leq x \leq 1
    • 0≤y≤x0 \leq y \leq x
    • 0≤z≤y0 \leq z \leq y
  • Set up the triple integral using the limits of integration and evaluate using iterated integration
    • ∭Rf(x,y,z)dV=∫01∫0x∫0yf(x,y,z)dzdydx\iiint_R f(x, y, z) dV = \int_0^1 \int_0^x \int_0^y f(x, y, z) dz dy dx
  • The of a triple integral is the solid region over which the integration is performed

Order optimization in triple integrals

  • Analyze the region RR and the function f(x,y,z)f(x, y, z) to determine the most convenient order of integration that simplifies the limits of integration or the integrand
  • allows changing the order of integration without affecting the value of the triple integral for continuous functions over a closed, bounded region RR
    • ∭Rf(x,y,z)dV=∫ab∫cd∫eff(x,y,z)dzdydx=∫cd∫ab∫eff(x,y,z)dzdxdy\iiint_R f(x, y, z) dV = \int_a^b \int_c^d \int_e^f f(x, y, z) dz dy dx = \int_c^d \int_a^b \int_e^f f(x, y, z) dz dx dy
  • Changing the order of integration can simplify the integrand (xy+zxy + z becomes a constant when integrating with respect to zz first)

Average value using triple integrals

  • The average value fˉ\bar{f} of a function f(x,y,z)f(x, y, z) over a closed, bounded region RR is given by:
    • fˉ=∭Rf(x,y,z)dV∭RdV\bar{f} = \frac{\iiint_R f(x, y, z) dV}{\iiint_R dV}
  • Compute two triple integrals to find the average value:
    1. The triple integral of the function f(x,y,z)f(x, y, z) over the region RR
    2. The triple integral of the constant function 11 over the region RR, giving the volume of RR
  • Divide the first integral by the second integral to obtain the average value
  • The average value represents the balance point or center of mass of the solid represented by f(x,y,z)f(x, y, z) over RR, with applications in physics and engineering

Applications of Triple Integrals

  • : Triple integrals can be used to compute the volume of in three-dimensional space
  • Partial derivatives: The evaluation of triple integrals often involves taking partial derivatives with respect to the variables of integration

Key Terms to Review (20)

∭: The symbol ∭ represents a triple integral, which is used to compute the volume under a surface in three-dimensional space. Triple integrals allow for the evaluation of functions of three variables over a specified region, enabling the calculation of volumes, mass, and other properties of three-dimensional objects. This concept extends the idea of integration from single and double integrals into a third dimension, accommodating more complex shapes and volumes.
Cartesian Coordinates: Cartesian coordinates are a system used to locate points in space by specifying their positions along orthogonal (perpendicular) axes. This coordinate system provides a way to describe the location of objects in two-dimensional (2D) or three-dimensional (3D) space using numerical values.
Change of Variables: Change of variables is a mathematical technique used to transform an integral from one set of variables to another. This transformation allows for simplification and evaluation of integrals that would otherwise be difficult or impossible to solve in their original form.
Cuboid: A cuboid is a three-dimensional geometric figure bounded by six rectangular faces, where opposite faces are parallel and congruent. It can also be described as a rectangular prism, and its volume can be calculated using the formula $$V = l imes w imes h$$, where $$l$$ is the length, $$w$$ is the width, and $$h$$ is the height. In the context of triple integrals, cuboids are essential for defining the integration bounds when calculating volumes in three-dimensional space.
Cylinder: A cylinder is a three-dimensional geometric shape that is formed by the rotation of a rectangle around one of its sides. It has a circular base and a curved surface that connects the two parallel circular bases.
Cylindrical Coordinates: Cylindrical coordinates are an alternative coordinate system used to describe the position of a point in three-dimensional space. Unlike the traditional Cartesian coordinate system, cylindrical coordinates use a radial distance, an angle, and a height to uniquely identify a location, providing a more natural way to represent certain geometric shapes and physical phenomena.
Domain: The domain of a function is the set of all possible input values (or 'x' values) for which the function is defined. Understanding the domain is crucial because it helps identify where a function can be evaluated and influences its limits and continuity, as well as the regions over which integrals can be calculated in higher dimensions.
DV: In calculus, 'dV' represents an infinitesimal volume element used in the context of triple integrals. It helps to break down three-dimensional regions into smaller, manageable parts for integration. This concept is essential when calculating volumes or integrating functions over a three-dimensional space, and it can take different forms depending on the coordinate system being used, such as Cartesian, cylindrical, or spherical coordinates.
Fubini's theorem: Fubini's theorem is a fundamental principle in calculus that allows the evaluation of double integrals by iteratively integrating with respect to one variable at a time. This theorem establishes that if a function is continuous on a rectangular region, then the double integral can be computed as an iterated integral, making it possible to switch the order of integration without changing the value of the integral.
Iterated Integral: An iterated integral is a method of evaluating multiple integrals by breaking them down into successive integrations, typically applied to functions of two or three variables. This approach allows the computation of double and triple integrals by integrating one variable at a time while treating other variables as constants, which simplifies the evaluation process. It is particularly useful when dealing with complex regions or functions, enabling calculations over rectangular and general regions in both two and three dimensions.
Jacobian: The Jacobian is a matrix that represents the rates of change of a vector-valued function with respect to its variables, capturing how the output changes as the input varies. This concept is crucial for transformations in multiple integrals, especially when changing from one coordinate system to another, ensuring accurate calculation of area and volume.
Multiple Integration: Multiple integration refers to the process of evaluating integrals with more than one variable. It involves the sequential application of single integrals to compute the total integral over a multi-dimensional region. This concept is central to the topics of 'Double Integrals over Rectangular Regions' and 'Triple Integrals' in calculus.
Order of Integration: The order of integration refers to the sequence in which the integration variables are evaluated when performing multiple integrals, such as double integrals or triple integrals. The order of integration determines the structure and evaluation of the integral, and it can significantly impact the final result.
Partial Derivatives: Partial derivatives are a type of derivative that measure the rate of change of a multivariable function with respect to one of its variables, while treating the other variables as constants. They provide a way to analyze the sensitivity of a function to changes in its individual inputs.
Solid Regions: Solid regions refer to the three-dimensional areas in space where a function is defined and integrated over. They are essential in understanding triple integrals, as these regions define the boundaries within which calculations occur, allowing for the evaluation of volume and mass among other properties.
Sphere: A sphere is a three-dimensional geometric shape that is perfectly round, with all points on its surface equidistant from the center. It is one of the most fundamental shapes in mathematics and has numerous applications in various fields, including calculus, physics, and engineering.
Spherical Coordinates: Spherical coordinates are a three-dimensional coordinate system that uses three values, $r$, $\theta$, and $\phi$, to specify the location of a point in space. This system provides a natural way to describe positions on the surface of a sphere or within a spherical volume, and is widely used in various fields of mathematics, physics, and engineering.
Triple Integral: A triple integral is a three-dimensional extension of the definite integral, used to calculate the volume of a three-dimensional region or to integrate a function over a three-dimensional domain. It is a fundamental concept in multivariable calculus and is closely related to the understanding of limits, continuity, and changes of variables in multiple integrals.
Volume calculation: Volume calculation involves determining the amount of three-dimensional space that an object occupies, often represented in cubic units. This concept is crucial when using triple integrals, as it allows for the computation of volumes for various shapes and regions in three-dimensional space by integrating a function over a specified domain.
Volume Element: The volume element, also known as the differential volume, is a fundamental concept in multiple integral calculus that represents an infinitesimally small volume within a larger three-dimensional region. This volume element is a crucial component in the evaluation of triple integrals, as well as in the transformation of integrals from one coordinate system to another, such as cylindrical or spherical coordinates.
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