Problem 1 - University Calculus: Early Transcendentals

Question

Sketch the described regions of integration. $$0 \leq x \leq 3, \quad 0 \leq y \leq 2 x$$

Step-by-Step Solution

Answer

The region is a triangle with vertices at (0, 0), (3, 0), and (3, 6).

Step 1: Understand the Region Description

The given region is defined by the inequalities: \(0 \leq x \leq 3\) and \(0 \leq y \leq 2x\). This means \(x\) ranges from 0 to 3, and for each \(x\), \(y\) ranges from 0 to \(2x\).

Step 2: Determine Boundary Lines

Consider the boundary conditions: \(x = 0\), \(x = 3\), and \(y = 2x\). The line \(y = 2x\) is a straight line starting from the origin (0, 0) and having a slope of 2.

Step 3: Sketch the Region

Start by sketching the axes. Plot the line \(x = 3\), which is a vertical line parallel to the y-axis passing through x=3. Next, plot \(y = 2x\), starting from the origin and passing through the point (3, 6) since at \(x = 3\), \(y = 2(3) = 6\). The region is bounded by \(y = 2x\), the x-axis, and the line \(x = 3\).

Step 4: Identify the Bounded Area

Highlight the triangular region which lies below the line \(y = 2x\), to the right of the y-axis (\(x=0\)), and to the left of the line \(x=3\). This region forms a right triangle with vertices at (0, 0), (3, 0), and (3, 6).

Key Concepts

Boundary Conditions

Boundary conditions are critical in defining the limits of a region of integration. They determine where the region starts, stops, and outline its limits.
In our exercise, we have two key conditions:

• The range for \(x\) is from 0 to 3. This tells us that the region spans horizontally from the y-axis until it meets the vertical line at \(x=3\).
• The range for \(y\) is from the x-axis (where \(y=0\)) up to the line \(y=2x\). This line dictates the top boundary of our region.

These boundary conditions help form the geographical edges of the area we are examining.

Inequalities

The inequalities \( 0 \leq x \leq 3 \) and \( 0 \leq y \leq 2x \) describe a range of possible values for \(x\) and \(y\).
They tell us something very important:

• \(x\) can be any value between 0 and 3, inclusive.
• For any chosen value of \(x\), \(y\) can vary from 0 to double the \(x\) value (since it goes up to \(2x\)).

By keeping within these inequalities, we can ensure that every point we consider belongs to the desired region of integration.

Sketching Regions

To visualize a region of integration, sketching is a crucial skill. It transforms abstract inequalities into a visible shape.
Here's how you can tackle it:

• Start by drawing the coordinate axes.
• Mark the vertical line \(x=3\), which sets one side of our region.
• Next, draw the line \(y=2x\), which starts at the origin and passes through the point (3,6).
• Finally, the region consists of the area above \(x=0\), below the line \(y=2x\), and to the left of \(x=3\).

With these steps, the sketched area should give a clearer picture of the region's boundaries and limits.

Right Triangle

This exercise identifies a region that forms a right triangle, which is one of the simplest geometrical shapes.
A right triangle has one 90-degree angle, with two sides perpendicular to each other.

• In this case, we have vertices at (0,0), (3,0), and (3,6).
• The line \(x=3\) acts as one leg of the triangle, and the segment along the x-axis from (0,0) to (3,0) acts as the other leg.
• The hypotenuse is represented by the line \(y=2x\) from (0,0) to (3,6).

Understanding this triangle helps delineate the specific area under consideration and simplifies the integration process.