Rectangular triple integral

Definition

A rectangular triple integral is a mathematical operation used to compute the volume under a surface in three-dimensional space over a rectangular region. This integral is evaluated as an iterated integral, where the volume is calculated by integrating a function three times, typically with respect to the variables x, y, and z, each over specified intervals that define a rectangular box in 3D space.

5 Must Know Facts For Your Next Test

1. To evaluate a rectangular triple integral, you perform three successive integrations, often written as $$\int_{a}^{b} \int_{c}^{d} \int_{e}^{f} f(x,y,z) \, dz \, dy \, dx$$.
2. The order of integration can be changed without affecting the result if the limits are constants, but care must be taken when dealing with variable limits.
3. When calculating a rectangular triple integral, the function being integrated must be continuous over the specified rectangular region to ensure accurate results.
4. Rectangular triple integrals can be applied in various fields such as physics and engineering to find quantities like mass, charge, or probability distributed in three-dimensional space.
5. To visualize a rectangular triple integral, imagine slicing a three-dimensional box into infinitesimally small elements, calculating the volume of each element, and summing them up through integration.

Review Questions

• How do you set up a rectangular triple integral for a given function and rectangular region?
  • To set up a rectangular triple integral, first identify the function you want to integrate and determine the limits for each variable that define your rectangular region. The integral will be structured as an iterated integral: $$\int_{a}^{b} \int_{c}^{d} \int_{e}^{f} f(x,y,z) \, dz \, dy \, dx$$. Make sure to establish the correct order of integration based on how the limits relate to each other.
• Discuss how changing the order of integration in a rectangular triple integral can affect its evaluation.
  • Changing the order of integration in a rectangular triple integral can simplify calculations or make them more complex depending on the integrand and the limits. If all limits are constants, switching the order does not change the value of the integral. However, if there are variable limits dependent on other variables, you must carefully adjust those limits when changing their order to ensure accuracy in calculating the volume.
• Evaluate a specific rectangular triple integral over a defined region and explain your process step by step.
  • To evaluate a specific rectangular triple integral, first write out the function and identify your limits for x, y, and z. For instance, if evaluating $$\int_{0}^{1} \int_{0}^{2} \int_{0}^{3} (xyz) \, dz \, dy \, dx$$, start with the innermost integral with respect to z. Solve $$\int_{0}^{3} (xyz) \, dz$$ first, then substitute that result into the next integral with respect to y: $$\int_{0}^{2} (result) \, dy$$. Finally, take that outcome and integrate with respect to x: $$\int_{0}^{1} (result) \, dx$$. Each step involves plugging in limits and simplifying until you arrive at a final numerical answer.

Related terms

iterated integral: An iterated integral is the process of evaluating multiple integrals in succession, where the result of one integral becomes part of the next integration process.

volume: Volume is the measure of the amount of space an object occupies, which can be calculated using integrals when dealing with irregular shapes.

limits of integration: Limits of integration are the bounds that specify the interval over which an integral is computed, determining the range of values for the variables involved.