Problem 34 - University Calculus: Early Transcendentals

Question

Sketch the region of integration and write an equivalent double integral with the order of integration reversed. $$\int_{0}^{2} \int_{y-2}^{0} d x d y$$

Step-by-Step Solution

Answer

The reversed order integral is \( \int_{-2}^{0} \int_{x+2}^{2} dy \, dx \).

Step 1: Understanding the Original Order of Integration

In the given integral, the order of integration is dy dx. The limits of integration imply that x varies from y-2 to 0 and y ranges from 0 to 2. The integral describes a region in the xy-plane.

Step 2: Sketching the Region of Integration

To sketch the region, note that for each value of y, x varies from y-2 to 0. Start by sketching the line x = y-2. This line intersects the x-axis at (0, -2) and the y-axis at (2, 0). The upper boundary for x is the y-axis (x = 0). The region of integration is hence bounded by x = y-2, x = 0, y = 0, and y = 2.

Step 3: Determine New Limits for x

The region can also be described by varying x. For any x in this region, y must be greater than or equal to x + 2. Also, y should be less than or equal to 2. Thus, for a given x, y ranges from x + 2 to 2.

Step 4: Determine New Limits for y

As x goes from line x = y-2 to x=0 and y from 0 to 2 in the original integration, the bounds of x vary from -2 to 0.

Step 5: Write the New Double Integral

The new order of integration, dx dy, changes the limits: y varies from x + 2 to 2 and x varies from -2 to 0. Thus, the equivalent double integral with reversed order is \[\int_{-2}^{0} \int_{x+2}^{2} dy \, dx \]

Key Concepts

Region of Integration

When we deal with double integrals, it's crucial to understand the region of integration, as it helps define what part of the plane we're interested in. This region is the area in the xy-plane over which the integration takes place. For the given integral \[ \int_{0}^{2} \int_{y-2}^{0} dx \, dy \]we determined that the region is bounded by:

• The line \(x = y-2\), which intersects the axes at (0, -2) and (2, 0).
• The line \(x = 0\), which represents the y-axis.
• The horizontal lines \(y = 0\) and \(y = 2\).

Sketching this region involves plotting these lines and shading the area bounded by them. In simpler terms, envision a slice of the xy-plane framed by these boundaries. This slice tells us where the integration sums the values of the function.

Order of Integration

In a double integral, the order of integration refers to the sequence in which integration is performed, typically represented as \(\int \int dx \, dy\) or \(\int \int dy \, dx\).It's the difference between integrating with respect to x first and y second, or vice versa.In the original exercise, the order of integration is \( dx \, dy \).This means you integrate with respect to x first while treating y as a constant, and then you integrate the result with respect to y.Reversing this order means switching these variables' roles in the integration process. By altering it to \( dy \, dx \), each x corresponds to y-values that directly relate to x plus a constant. Understanding this concept allows us to manipulate integrals for easier computation or to describe the same region of integration in different ways.

Limits of Integration

The limits of integration specify the boundaries over which to integrate for each variable. They are defined by inequalities that relate to the region of integration. Initially, the limits were:

• \(y\) ranging from 0 to 2
• \(x\) ranging from \(y-2\) to 0 for each fixed \(y\)

These limits describe a specific region based on the y variable's values. Upon reversing the order of integration, the limits change to:

• \(x\) ranging from -2 to 0
• \(y\) ranging from \(x+2\) to 2 for each fixed \(x\)

This new set of limits corresponds to integrating with respect to y first and then x. Calculating the double integral with these limits involves "scanning" the region in the opposite direction, tracking x's boundaries first, and subsequently y's extensions, keeping it within the region we initially defined.