3. In three-dimensional space, a standard coordinate system is given. Find the area of the set of points satisfying the following conditions: $x^{2}+y^{2}=5,|x-y|<1,|y-z|<1$.
Solution. We will construct the set of points in the $xy$-plane satisfying the first two conditions (which do not depend on $z$). The first condition defines a circle, the second - a strip, their intersection is two arcs. The length of each arc is $\sqrt{5}(\operatorname{arctg} 2-\operatorname{arctg} 1 / 2)$.
![](https://cdn.mathpix.com/cropped/2024_05_06_2b7c43278745f884d641g-1.jpg?height=480&width=1522&top_left_y=1411&top_left_x=298)

In three-dimensional space, the circle corresponds to a cylindrical surface, and the arcs to two strips on it. The figure on the right shows the development of this surface; the strips satisfying the second condition are shaded blue. The region satisfying the third condition is shown in orange. This region is a strip of width 2, running along the line $z=y$. (This line is a sine wave; however, this fact is not used in the solution.) The figure of interest - the intersection of the two vertical strips with the "sine wave" - consists of two equal parts. The area of each part is twice the width of the blue strip, which is easy to establish by rearranging the curvilinear triangle, as shown in the last figure. And the width of the blue strip is the length of the arc found earlier.

Answer: $4 \sqrt{5}(\operatorname{arctg} 2-\operatorname{arctg} 1 / 2)$.