Question

Write the given iterated integral as an iterated integral with the indicated order of integration. \(\int_{0}^{2} \int_{0}^{4-2 y} \int_{0}^{4-2 y-z} f(x, y, z) d x d z d y ; d z d y d x\)

Short answer

Reordered integral: \(\int_{0}^{2} \int_{0}^{4-2y-z} \int_{0}^{4-2y} f(x, y, z) \ dz \ dy \ dx\).

Step-by-step solution

Understand the Original Integral

The original iterated integral is given as \( \int_{0}^{2} \int_{0}^{4-2 y} \int_{0}^{4-2 y-z} f(x, y, z) dx \, dz \, dy \). This means we first integrate with respect to \(x\), then with respect to \(z\), and finally with respect to \(y\).

Analyze the Order of Integration

We need to rearrange the order of integration to \(dz \, dy \, dx\). This requires adjusting the limits of integration to match this new order.

Determine New Limits for \(dz\)

From the original setup, \(x\) goes from \(0\) to \(4-2y-z\), hence \(z\) was previously limited by \(0 \leq z \leq 4-2y\). However, considering the order \(dz\) first, \(z\) must now adjust based on the new limits of integration where \(x\) is already known to be from \(0\) to \(2\).

Determine New Limits for \(dy\)

Since originally \(y\) limits were outermost, indicating \(y\) from \(0\) to \(2\). For \(dy\) to be in the middle, it remains as \(0\) to \(2\) as it covers the entire region. However, the interdependence will adjust the integration boundaries indirectly as \(z\) limits vary.

Determine New Limits for \(dx\)

As \(dx\) is now innermost, this means \(x\) integrates directly from limits unaffected by outer integration of \(z\) and \(y\). So, \(x\) varies from \(0\) to \(4-2y-z\) ensuring \(x\) is adapted within \(dz \, dy\).

Rewrite the Integral

Rewriting the integral with new integration order and limits is:\[\int_{0}^{2} \int_{0}^{4-2y-z} \int_{0}^{4-2y} f(x, y, z) \ dz \ dy \ dx\]

Key concepts

Order of Integration

In iterated integrals, the order of integration refers to the sequence in which integration is performed along different variables. The process involves choosing which variable to integrate first, second, and third when dealing with three-variable functions. The order is important because it can affect the difficulty of computation and the resulting integrals.For example, consider an iterated integral written as \( \int \int \int f(x, y, z) \, dx \, dz \, dy \), which means you integrate with respect to \(x\) first, then \(z\), then \(y\). However, if the order is changed to \( \int \int \int f(x, y, z) \, dz \, dy \, dx \), you would start with integration with respect to \(z\), followed by \(y\), and finally \(x\).Changing the order of integration often requires adjusting the limits of integration to accommodate intertwined variable dependencies, which can simplify calculations or conform to geometric constraints. Understanding the various ways to approach the problem can enhance the intuition for solving more complex multivariable calculus problems.

Limits of Integration

Limits of integration in iterated integrals define the range over which we integrate each variable. This can be thought of as setting the bounds or boundaries for integrating along these variables. Depending on the function's context and the order of integration, these limits can change.Each limit specifies where one variable starts and ends while another variable is held fixed, or varies within certain constraints. For instance, if you start by integrating with respect to \(x\), the limits might be specific numeric boundaries such as \(0 \leq x \leq 4-2y-z\). However, when the order is rearranged, these limits must often be recalculated. To correctly reestablish the limits, we examine the geometric or algebraic relationships stated by the problem. In some scenarios, limits could be constants (like \(0\) to \(2\)), while in others, they depend on functions of the other variables.

Multivariable Calculus

Multivariable calculus extends basic calculus concepts to functions of more than one variable, significantly increasing the versatility and applications of mathematical models in real-world scenarios. In this branch of calculus, we often deal with functions like \(f(x, y, z)\) that have multiple inputs and outputs, managing longer chains of operations and dependencies.In problems involving iterated integrals, multivariable calculus allows us to explore the nature of three-dimensional space by evaluating complex integrals across multiple dimensions. This is widely useful in physics, engineering, and higher mathematics where we must take into account the interaction between several continuously varying factors.A fundamental part of multivariable calculus is understanding the geometrical interpretation of the functions and the regions over which they are integrated. Different orders or methods of integration are applied depending on these geometric constraints and the specific nature of the problem being solved.