Problem 4 - University Calculus: Early Transcendentals

Question

Sketch the described regions of integration. $$0 \leq y \leq 1, \quad y \leq x \leq 2 y$$

Step-by-Step Solution

Answer

The region is a trapezoid bounded by \(x = y\), \(x = 2y\), \(y = 0\), and \(y = 1\).

Step 1: Identify the Region from Inequalities

First, identify the region defined by the inequalities. The inequality \(0 \leq y \leq 1\) describes a horizontal band stretching from \(y = 0\) to \(y = 1\). The inequalities \(y \leq x \leq 2y\) place the region between the two lines \(x = y\) and \(x = 2y\).

Step 2: Analyze the Overlapping Regions

To find the valid intersection, consider the region where the x-range curves meet the y-range. The line \(x = y\) intersects with the horizontal band as \(0 \leq y \leq 1\) when \(x = y\). Similarly, the line \(x = 2y\) intersects from \(x = 0\) to \(x = 2\) at \(y = 1\).

Step 3: Draw the Boundary Lines

Draw the boundary lines \(y = 0\) and \(y = 1\) horizontally, representing the limits for \(y\). Then draw the lines \(x = y\) and \(x = 2y\). These lines will intersect the allowed \(y\) values within the specified range.

Step 4: Visualize the Region of Integration

The region of integration is trapezoidal. It is bounded by the lines \(y = 0\), \(y = 1\), \(x = y\), and \(x = 2y\). It starts at the point (0,0) where \(y=0\) and \(x=y\), and it ends at the point (2,1) where \(y=1\) and \(x=2y\).

Step 5: Sketch the Region of Integration

Draw the plot with a coordinate plane. First, draw the horizontal line for \(y = 0\) and \(y = 1\). Then, plot the lines \(x = y\) and \(x = 2y\), ensuring they intersect at correct points. Shade the area where all conditions are true, resulting in a trapezoid shape.

Key Concepts

Region of Integration

A region of integration in integral calculus is essentially the area of interest over which you will be integrating a function. In our exercise, the region is determined by given inequalities, focusing on coordinates \(x\) and \(y\). Specifically:

• The inequality \(0 \leq y \leq 1\) defines a horizontal strip extending from \(y = 0\) to \(y = 1\). This strip serves as our range for \(y\).
• Meanwhile, the inequalities \(y \leq x \leq 2y\) overlay this strip, delineating a space between the lines \(x = y\) and \(x = 2y\).

To understand this region, you must visualize the intersection of these conditions. Begin by sketching the lines that set the boundaries: horizontal lines at \(y = 0\) and \(y = 1\), and sloped lines for \(x = y\) and \(x = 2y\). The trapezoidal area confined within these lines is the region of integration. This concept is fundamental as it guides you in setting the limits for multiple integrals.

Integral Calculus

Integral Calculus is a branch of calculus focused on the concept of integration. Integration is the process of finding the whole given its parts, like finding the area under a curve.In multiple integration, as highlighted in our problem, we integrate over more than one variable. Here, \(x\) and \(y\) variables define the region on a two-dimensional plane. Multiple integrals are used to compute areas, volumes, and other quantities that accrue over a region.The region of integration is crucial for setting up such integrals. By clearly defining the boundaries (\(0 \leq y \leq 1\) and \(y \leq x \leq 2y\)), we are equipped to apply either double or iterated integrals. These integrals help calculate values of functions over the defined area. To solve this step easily, visualize how these boundaries translate onto a graph and create an enclosed region for integration. Drawing aids in conceptualizing the flow of integration across this visual representation.

Coordinate Geometry

Coordinate Geometry, or analytic geometry, brings algebra and geometry together. It allows us to prove geometric theorems and find properties of shapes using algebraic equations.In this exercise, we apply coordinate geometry to delineate the region of integration. By understanding the equations \(x = y\) and \(x = 2y\), we recognize them as straight lines on a coordinate plane:

• \(x = y\) is a line at 45 degrees, where any point on the line has equal \(x\) and \(y\) coordinates.
• \(x = 2y\) represents a line where \(x\) is twice the \(y\) value, effectively steepening the slope compared to \(x = y\).

The intersection points of these lines with the horizontal boundaries \(y = 0\) and \(y = 1\) help in completing the bounded trapezoidal shape. The interplay of these lines in coordinate geometry illustrates the space where the function integrates, reinforcing how linear boundaries guide the integration process.