Skizzieren Sie den Rotationskörper, der durch Drehung der Kurve \(z=\sqrt{4-x^{2}}\) um die \(z\)-Achse entsteht. Wie lautet die Funktionsgleichung der Rotationsfld?che, fur welche Wertepaare \((x ; y)\) ist diese Funktion definiert? Bestimmen Sie ferner die Schnittkurven der Fl?che mit den drei Koordinatenebenen (Schnittkurvendiagramme).

A rotation body, also known as a solid of revolution, is a three-dimensional object obtained by rotating a two-dimensional curve around a specified axis.
In our given exercise, we start with a curve defined by the equation \( z = \sqrt{4 - x^{2}} \).
This curve forms a semicircle in the xz-plane. To create the rotation body, we rotate this curve around the z-axis.
The result is a three-dimensional shape that encompasses all the points generated during the rotation.
Understanding how curves form shapes through rotation is crucial in many areas, such as physics and engineering.

A surface of revolution is created when a two-dimensional curve is rotated around an axis. The equation defining our original curve is \( z = \sqrt{4 - x^{2}} \).
When this semicircle is rotated around the z-axis, the resulting surface can be described by the equation \( z = \sqrt{4 - x^{2} - y^{2}} \).
This new equation includes both x and y variables and represents the outer shell of the rotation body.
The surface of revolution is smooth and symmetric around the z-axis, resembling a sphere with a removed portion.
This surface is key to understanding the entire three-dimensional shape formed by the rotation.

Intersection curves are the lines formed where our 3D surface meets the coordinate planes (xy, xz, and yz planes).
Finding these curves helps us visualize and understand the shape better.

1. In the **xy-plane (z = 0)**, we set \( z = 0 \) in \( z = \sqrt{4 - x^{2} - y^{2}} \), obtaining \( x^{2} + y^{2} = 4 \). This is a circle with radius 2 centered at the origin.

2. In the **xz-plane (y = 0)**, we set \( y = 0 \), giving the original equation \( z = \sqrt{4 - x^{2}} \). This represents a semicircle of radius 2 centered at the origin in the xz-plane.

3. For the **yz-plane (x = 0)**, setting \( x = 0 \), we get \( z = \sqrt{4 - y^{2}} \). This is another semicircle of radius 2 centered at the origin in the yz-plane.
By analyzing these intersection curves, we build a fuller understanding of the overall shape and size of the rotation body.

The domain of a function includes all the possible input values (x, y) that allow the function to produce a real number output.
For our surface, the function is defined as \( z = \sqrt{4 - x^{2} - y^{2}} \).
To ensure the argument under the square root is non-negative, we solve: \( 4 - x^{2} - y^{2} \ge 0 \).
This simplifies to \( x^{2} + y^{2} \le 4 \), representing a disk of radius 2 in the xy-plane, centered at the origin.
Thus, the function is only defined within this circular region.
Understanding the domain is essential for plotting and working with the surface in practical applications.

A semicircle is half of a circle.
In the exercise, the curve \( z = \sqrt{4 - x^{2}} \) forms a semicircle in the xz-plane.
This is because for each value of x between -2 and 2, z ranges from 0 to 2, covering only the upper half of the circle.
When we rotate this semicircle around the z-axis, we form a surface of revolution described by \( z = \sqrt{4 - x^{2} - y^{2}} \).
Understanding the semicircle helps us visualize the initial shape before rotation and appreciate the full three-dimensional geometry created afterward.