It seems to me that you might have the misconception that high-dimensional calculus or linear algebra may have dubious practical value because the world around us looks 3 (or 4) dimensional. The crucial point being misunderstood by this impression is that the main reason high-dimensional math arises in practical problems is because the problems involve many parameters. Each new parameter is often a new coordinate, or is its own dimension (degree of freedom).
For example, in computational chemistry the description of a molecule with many atoms is described by many coordinates at once: 3 each for the position of each atom, extra coordinates to describe the lengths of angles of chemical bonds, etc.
Another example of a practical problem where the framework for thinking about it uses very high dimensions is developing a recommendation system (like Netflix suggesting movies you might like). The description makes each user a separate dimension: that's a space with millions of dimensions! I'm not saying volume calculations are made here, but it's important to appreciate why high-dimensional settings show up first.
You want to know practical reasons for caring about high-dimensional volumes. If you need to integrate in high dimensions then you need a language to discuss volumes in high dimensions. High dimensional integration occurs in many areas, such as mathematical finance (see here), statistics (see here), and statistical mechanics (see here).