Chapter 13. Boxes
Integrals are tough. Integrals make us work hard to do something simple: calculate a length or area. In its simplest form, an integral tells us that the length of the pictured interval is three, or the area of the box is six.
Given that we can already easily find such lengths and areas without using integrals, one might get to wondering, why should we learn about integrals?
Integrals are very powerful tools for measuring: we can switch out a standard ruler \( \bar{\standard}_{x} \) for a non-standard ruler \( \bar{\diff} z : \Ruler_{} \) that varies from point to point. A non-standard ruler can give more weight to one location than another. We might picture a non-standard ruler as having a non-uniform density of markings.
Non-standard rulers appear in any number of contexts: by changing coordinates, we'll turn standard rulers into non-standard rulers. We'll find that non-standard rulers can be useful even if our ultimate measurement involves only standard rulers!