Question

$$\text { Sketch the region } D=\left\\{(x, y, z): x^{2}+y^{2} \leq 4,0 \leq z \leq 4\right\\}$$

Short answer

Question: Sketch the region D in three-dimensional space defined by the inequalities: \(x^2 + y^2 \leq 4\) and \(0 \leq z \leq 4\).

Step-by-step solution

Draw the circle

Start by sketching the circle in the xy-plane, defined by the equation \(x^2 + y^2 = 4\). This circle has a center at the origin (0,0) and a radius of 2 units. Draw the circle using a compass or freehand.

Determine the height

From the inequality \(0 \leq z \leq 4\), we understand that the height of the region extends from 0 to 4 units in the z-direction. In our sketch, we will denote the z-axis vertically above the circle.

Extrude the circle vertically

Now that we have the circle in the xy-plane and know the height of our region, we will extrude the circle vertically along the z-axis for a distance of 4 units. To do this, draw two parallel lines, each 4 units in length, starting from the edge of the circle and extending upward. The top endpoints of these lines should be connected to form another circle with the same radius as the one in the xy-plane.

Connect the top and bottom circles

After drawing the top circle, connect the corresponding points of the top and bottom circles with straight lines to represent the vertical sides of the region.

Label the region

Finally, label the region as D inside the cylindrical shape and label each of the axes as x, y, and z. The completed sketch represents the region D defined by the inequalities: \(x^2 + y^2 \leq 4\) and \(0 \leq z \leq 4\).

Key concepts

Understanding the xy-plane

The xy-plane is a fundamental concept in geometry and mathematics. It is a two-dimensional flat surface where any point can be identified with just two coordinates, x and y. Think of it like a grid or a chessboard where we can plot points just by knowing their position in the horizontal (x) and vertical (y) directions.

When talking about the xy-plane, you might hear terms like the x-axis or the y-axis. These are simply lines that cross each other at right angles, creating the horizontal and vertical dimensions of the plane, respectively. The point where they meet is called the origin, labeled as (0, 0).

• **x-coordinate**: tells you how far left or right the point is.
• **y-coordinate**: tells you how far up or down the point is.

In our exercise, the circle we sketched belongs to this xy-plane. The equation \(x^2 + y^2 = 4\) describes a circle with its center at the origin and a radius of 2 units. This equation holds because each point on this circle is exactly 2 units away from the center.

Understanding inequalities in geometry

Inequalities are mathematical expressions involving the symbols \(<, >, \leq, \geq\), which help define boundaries and constraints within a problem. They allow us to specify ranges where certain values can exist, such as ensuring one number is greater than another or within a certain range.

In the context of the exercise, the first inequality \(x^2 + y^2 \leq 4\) limits our region to points within and on the circle in the xy-plane. This means any point (x, y) must lie inside or on the edge of the circle of radius 2.

• Circle equation: \(x^2 + y^2 = 4\)
• Inequality: \(x^2 + y^2 \leq 4\)

The second inequality \(0 \leq z \leq 4\) restricts our z-values between 0 and 4. This tells us about the height of the region extending vertically. Think of it as setting limits for how high or low a point can be in the space around the circle we've drawn.

Exploring three-dimensional geometry

Three-dimensional geometry involves objects that have length, width, and height. Unlike two-dimensional shapes, these forms occupy space around us, such as cubes and spheres.

In our exercise, we're looking at a cylindrical shape that has emerged from the circle in the xy-plane. This is achieved by extruding—or stretching—the circle upwards to form a complete three-dimensional object with both top and bottom circles.

• **Base (xy-plane circle)**: This is where our shape starts, as drawn from the equation \(x^2 + y^2 \leq 4\).
• **Height**: Given by the inequality \(0 \leq z \leq 4\), suggesting how tall the cylinder is.

Understanding this concept helps visualize how simple two-dimensional shapes like circles can generate complex forms. In our case, by extending the circle vertically through a distance of 4 units, we create a cylinder—a common shape in three-dimensional geometry.