The principle behind the Integral Yoga advocated by Sri Aurobindo Ashram becomes very clear in the introduction chapter "Life and Yoga, Part I,Chapter I of Synthesis of Yoga". The chapter ends with note saying life is a yoga by itself, the constant evolution process of reaching the supreme spirit. Sri Aurobindo, points us to working of Mother Nature in this context. Everything in Mother Nature is constantly performing its duty without giving any thoughts to the consequence or the results of it.
Quoting Swamy Vivekanda's definition of Yoga, Yoga is a compression process of Human Being's evolution to few years, or at some cases few months, Sri Aurobindo explores the fundamentals behind the various Schools of Yoga known to human man kind. He explores the striking similarities between Science experiments to prove facts and yoga techniques. The aim of Science is to derive at hypothesis and subsequently prove or disprove the same by experiments. In chemistry or any kind of Process engineering sciences, existing or previously found elements or process are constantly reworked, either new assembled or disassembled to come up with new functionality.
Similarly Yoga also works constantly on the human body in the case of HathaYoga and Human Mind in the case of Rajayoga, and eventually make it more suitable for knowing the Supreme Divine. The nature has been constantly working on our body through various evolution process and now we are at a more advanced level. The next immediate job is similarly working on our mind and consciousness.
Hence the emphasis on yoga being constantly running in the background of our every activity, would fasten this evolution process, thus making this whole life a Yoga.