Problem 25 - Calculus an Applied Approach

Question

In Exercises \(25-32,\) sketch the region \(R\) whose area is given by the double integral. Then change the order of integration and show that both orders yield the same area. $$ \int_{0}^{1} \int_{0}^{2} d y d x $$

Step-by-Step Solution

Answer

The area of the region \(R\) is \(2\) square units for both orders of integration. This confirms the fact that changing the order of integration does not alter the result for any double integral, given that the region of integration remains the same.

Step 1: Graphing the Region

The region \(R\) is defined by the limits of integration. From the outer integral, \(x\) ranges from \(0\) to \(1\), and from the inner integral, \(y\) ranges from \(0\) to \(2\). This represents a rectangle in the xy-plane with vertices at \((0,0), (1,0), (1,2),\) and \((0,2)\)

Step 2: Original Order of Integration

Using the given order of integration, the double integral is computed as follows: \(\int_{0}^{1} \int_{0}^{2} d y d x = \int_{0}^{1}[y]_{0}^{2} dx = \int_{0}^{1} 2 dx = [2x]_{0}^{1} = 2(1) - 2(0) = 2\). Therefore, the area of the region \(R\) is \(2\) square units.

Step 3: Changing the Order of Integration

To reverse the order of integration, one needs to examine the region of integration. From the graph, it can be inferred that \(x\) ranges from \(0\) to \(1\) and \(y\) ranges from \(0\) to \(2\). Hence, the double integral becomes: \(\int_{0}^{2} \int_{0}^{1} d x d y = \int_{0}^{2}[x]_{0}^{1} dy = \int_{0}^{2} 1 dy = [y]_{0}^{2} = 2 - 0 = 2\). So, the area of the region \(R\) remains the same, \(2\) square units, even if the order of integration is changed.

Key Concepts

Changing Order of Integration

Understanding how to change the order of integration in a double integral is a key skill in multivariable calculus. It involves identifying the limits of integration for both variables and then reversing them. In our example, the original double integral was set up as \(\int_{0}^{1} \int_{0}^{2} dy dx\). To change the order, we need to graph the region and note the bounds for each variable. Once we understand the rectangular region where \(x\) ranges from 0 to 1 and \(y\) ranges from 0 to 2, we can swap the integrals to \(\int_{0}^{2} \int_{0}^{1} dx dy\). This process is crucial when we encounter integrals that are difficult to solve in their original setup or in applications where one order might be more practical than another. It's essential to grasp that changing the order of integration does not change the value of the integral when the region of integration is properly defined.

When you practice changing the order of integration, always check by computing the integrals both ways to confirm that they yield the same result. This reinforces your understanding and ensures accuracy in the problem-solving process.

Sketching Regions of Integration

Sketching the region of integration is an invaluable step in visualizing and solving double integrals. For our rectangle in the \(xy\)-plane, visualizing the area bounded by \(x=0\) to \(x=1\) and \(y=0\) to \(y=2\) simplifies the process of changing the order of integration. A clear sketch provides insight into whether the region is simple (like a rectangle or circle) or more complex (like an area bounded by curves).

To sketch a region effectively:

• Plot the bounds of integration on their respective axes.
• Connect these bounds to outline the region.
• Label the vertices and axes, and shade the area that represents the region of integration.

By practicing sketching, you enhance your spatial reasoning and reduce the risk of errors when setting up integrals, which can ultimately save time and improve accuracy.

Computing Double Integrals

Computing double integrals is a process that allows us to find the volume under a surface or the area of a region on the \(xy\)-plane. To compute a double integral, we iteratively integrate with respect to one variable while treating the other as a constant and then integrate the result with respect to the second variable.

In our example, we first integrated \(dy\), treating \(x\) as constant, and then integrated \(dx\). The computational steps involved finding the antiderivative of 1 with respect to \(y\), evaluating it from 0 to 2, and then integrating the result, which was a constant, with respect to \(x\) from 0 to 1. The key to computing double integrals efficiently is to:

• Fully understand the order of integration and the behavior of the variables within the given limits.
• Know the fundamental integration rules and techniques to find antiderivatives.
• Ensure accurate arithmetic during the evaluation of the bounds.

With practice, computing double integrals becomes a systematic and straightforward task.

Cartesian Plane Regions

Regions on the Cartesian plane can be used to establish the limits of integration for double integrals. These regions are defined by their boundaries, which can be lines, curves, or a combination of both. In the case of our example, the region was a simple rectangle. Regions can also be more complex, and understanding the nature of these regions is critical when setting up integrals.

To handle Cartesian plane regions, one should:

• Identify the type of region (rectangular, polar, etc.).
• Know the equations or inequalities that define the borders of the region.
• Distinguish between 'Type I' and 'Type II' regions, which indicate whether the region is better described by bounding \(x\) with functions of \(y\), or vice versa.

Through exposure to varying region types, students learn to adapt their approach to setting up integrals to match the complexities introduced by different shapes on the Cartesian plane.