he surface determined by the parametric equations x = z(cos u + u sin u), y = z(sin u − u cos u),0 ≤ u, z ≤ 1

The given parametric equations define a surface in three-dimensional space. Here's how you can understand it:

1. The parameters are u and z, both ranging from 0 to 1.

2. The x-coordinate of a point on the surface is given by z(cos u + u sin u). This means that the x-coordinate is a function of both u and z. As u and z vary from 0 to 1, the x-coordinate will change accordingly.

3. Similarly, the y-coordinate of a point on the surface is given by z(sin u - u cos u). This is also a function of both u and z.

4. The z-coordinate of a point on the surface is simply z, which varies from 0 to 1.

5. Therefore, as u and z vary from 0 to 1, they trace out a surface in three-dimensional space. The exact shape of the surface will depend on the specific forms of the functions in the x and y coordinates.

6. To visualize this surface, you could plug in the parametric equations into a graphing software and observe the shape that is formed.