Integrating kinematic density can give us a lot of information concerning the area and the length of a region in the plane. Here is an outline of useful integral formulas.

I. Points in the Intersection of Regions

Suppose we have two regions in the plane and ; these need not be convex. Let be the fixed region and let move. Suppose is a point in the plane. Let's find the measure of the set of points that lie in . If , then consider the integral

Recall from Problem 20 that, if P is fixed, . We have

where is the area of . We now have two expressions for , and we may equate them to find an expression for the measure of points that lie in the intersection of our two regions. (1)

This equality yields Figure 1.

II. Integral Formula using an element of Arc Length

Let and denote the area of and , respectively. These sets must both have boundaries whose lengths can be measured, called rectifiable curves. Let and be the lengths of the boundaries of and . Choose a point on , the boundary of , and call it , where is the element of arc length. Consider the following integral:

Similar to the method above, we will fix both components in order to find a system of equations. Then we have

where is the length of the arc of contained in . We solve this system to get

Because kinematic density is invariant under rigid motions, an analogous result follows.

If we add the last two equations together we get an expression that includes , the length of the boundary of . Finally

III. Multiple Convex Sets

Let , , and be three convex sets and let be the fixed set, while and are allowed to move. Their kinematic densities are and , respectively. We want to find by using equations (1) and (2) as well as the fact that .

and are constants since they are not affected by

Then,

References: Santalo