If you understand the idea of multiple inputs with variable gain feeding a power amp directly, you understand what could typically referred to as an integrated amplifier. There is substantial flexibility in naming the various pieces of equipment which make up a system. Mostly because there are so many variations in how a system can be constructed. You have stumbled into the murkiest of waters with your question.

You must look back at the history of the consumer audio market to better understand how items acquired their names. Remember, at one point, all bread was brown and there were no specialty products used to make a sandwich without a crust.

You haven't mentioned the term "control amplifier", which is another term for "pre amp". Control amp qualifies as both an old and a new term. Before we began using low output magnetic phono cartridges the need arose to switch between inputs to the power amplifier. Although the selections were relatively few, the ability to conveniently control whether we were listening to the radio or the phonograph in our General Electric console was a necessity. Along with switching, the need to control the volume was evident once we moved to electrical amplification and were no longer using just a big horn attached to a cactus needle. So the "control amplifier" came into existence as electronic circuits were added in order to smooth the passage of the signals along to the power amplifier and let us control what we heard.

When magnetic phono cartridges and magnetic tape heads became popular after the second world war, the consumer need then became both conveniently switching between sources and boosting the level of those two relatively low output devices. In the days of only a few companies controlling what the consumer purchased, the idea of planned obsolescence hadn't yet been introduced. Instead we built upon what we already owned. We took our newly purchased automobile and added a hitch to connect with the travel trailer we were planning to buy. And so on. This meant "pre amplification" amplifiers, better turntables separated from the speaker box and, as a side note, the second channel tuner for the new stereo broadcasts, were sold as separate units which fed to your pre amp/control amplifier. This continued until the time when most everyone looked at the jumble of interconnecting wires and thought it would be a good idea to once again reduce the number of boxes. Pre amplifiers for phono and tape were absorbed into the control amplifier and buying a "pre amplifier" became the accepted terminology. The functions remained the same and the unit still controlled which source you listened to, how much volume you heard and whether you added some "tone" control to the sound. Look back at a few of the pre amplifiers from the 1950's to see the explosion of ideas which permeated the times. You'll see switching facilites for inputs we no longer employ and the ability to control which pre amplification curve was applied to the phono inputs. The popularity, for a short while, of open reel tape brought about the introduction of the tape input/output loop we now call "tape monitor", though it is now disappearing from the landscape.

During the first Golden Age of consumer audio, the "consumer" portion of the term was in full swing and the American public was consuming whatever the companies produced. Just as there were few companies producing complete speaker systems, the hobbyist bought and often assembled their own components to build a system which met their indivual needs. This meant buying separate components to suit both your budget and your desires. The introduction of the integrated amplifier (which is really what the full function consoles of the '30's and '40's were) gave way to the integrated receiver which took away one more box and combined the most popular sources into one unit without the jumble of interconnecting wires. Then someone noticed two things. First, if you had a tuner built into your receiver, you might not want to buy another, better tuner to add to your system. Of course, they hoped you would buy their tuner. To make the point more emphatically, they called your attention to the fact that as you added all of these sources, with their associated circuits and cables, you added the potential for noise and consequent signal loss to the system. Well, of course, no one wanted to loose some of what they had just purchased. So, in order to move the sensitive, low level circuitry of the phono pre amp away from the ever larger transformers of the ever higher powered amplifiers of the time, the system began to separate again as consumers were given the multiple choices of whose product they would buy to assemble the best sounding system available. This separation of pieces eventually led to the second Golden Age of consumer audio. This specialization of pieces and tasks led to the introduction of specialized cables to make each component sound its best.

While the cable wars have caused many divisions within the audio community, taking in the larger view of the consumer audio world would let us see everything remained essentially stable in this configuration until the advent of CD's in 1982. At that point the consumer was urged to remove the turntable from their system and use only a digital source, which would provide "perfect sound - forever". Gone was the need for a phono pre amplifier. Tapes had given up the inconvenience, and superior sound, of the open reel variety and had become small cassettes where the tape loop function was all but unnecessary. However, being the good consumers we are, there was still the need to control the switching of the ever expanding choice of sources and the various, assorted functions which gave us control over our destiny as audio consumers. So back to the control amplifier we went, though it was now more frequently referred to as a line level "pre amp". While appearing as an oxymoron, the line level "pre amplifier" is often a reality in that there is some small amount of gain developed in these circuits as the designers have, over the last fifty years, learned to manipulate the system to better control what goes in and what comes out. A bit of gain and a touch of buffering work wonders in reducing the noises that wander into our systems and make the system more flexible in its configuration. Thus giving us more choices, and, as consumers, we love choices.

Pre amps and source items can be isolated from the resonant field of the speakers by long cables (if you choose the correct components) which places the power amplifier behind the speakers. But, in our search for the smallest bit of information that might be lost through the system's inner workings, we know each connection point brings its own penality in signal loss. So back to the integrated amplifier we have gone recently to eliminate those potential sources of signal loss. Without separating the pre amplifier/control amplifier from the power amplifier we can eliminate the need for interconnecting cables and the designer can better control the known factors in designing the perfect component. Since the designer now controls the input and output impedances and voltages of an integrated amplifier, the design can be optimized for getting the best results possible from the whole circuit. Of course, buying an integrated amplifier places us at the mercy of the designer and we loose the flexibility we had when we bought separate components. If we want this many watts, we have to buy the control/pre amp attached which has this many inputs. If we want this many inputs, we have to buy this many watts. And, what happens if we change our minds in mid stream or decide we aren't hearing just exactly what we think needs to be there? Well, with an integrated amplifier, you're pretty much screwed. While most of today's integrated amplifiers of any lineage will provide pre amp outputs, disconnecting the power amp of the integrated means you're back to separates. Unless you decide to bi-amp your system which provides even more choices.

The short answer to all of this is, there is no great answer to your question. Yes, you can configure your system with a power amp that has an input and a passive volume control. By providing the correct source voltage and impedance you would have a system that works well though it is limited in flexibility. Adding switching gives more flexibilty but adds the potential for noise and signal loss. Wouldn't you rather move that control section away from the transformers of the power amp? Wouldn't it work better if you had the control amp close to your source and your listening position rather than over by the speakers? If you want to add a turntable to your system, it will have to be placed away from the speakers. Do you then run long cables from the turntable to the pre amp? Most cartridges don't like long cables. Not all pre amps like long cables. So you're back to making decisions about how to best configure your system. And there is no one single, right answer.

The flexibility of system building means you can buy the component(s) which suit your needs and whims. Depending on how you configure your system, one way might be better than another. But if you change your mind and want to add a component or move the system around, that configuration then might not work best.

And, here is the final answer to your question that I will provide. We still use preamps, whether integrated into one box, two or more, in order to make the system as flexible as you desire. But to get the most out of what you buy, you must obey some basic rules of system building. And no one can give you specifics of that until you begin to make decisions. Each choice you make will have benefits and down sides. Every decision you make will affect the next one after it and the one before it. Yes, you can do "that", but in doing "that" you will have to consider "this". By separating the components into their individual tasks we gain flexibility but possibly loose some signal quality. By piling everything together we keep the signal path short but loose flexibility and possibly add noise. To gain the flexibility of placement, we must know a few things about how the system works electrically. Choose poorly and you might have done better to buy a receiver.

I know you were probably wanting a A + B = C answer to your question, but there isn't one that I can see. Any explanation involves looking at your particular system configuration and deciding which approach works best in that instance. There is no real need for anything other than a way to control volume which is done by adjusting the voltage level going into the power amplifier. But how that is done most efficiently differs from system to system and listener to listener. Once you decide how you want your system put together, discuss the potential benefits and pitfalls of such a design with a competent retailer. They will tell you all about the choices you will have to make and the choices you will be forced into accepting by making "that" decision. Then possibly you will better understand why we have pre amplifiers or why you might not need one.