Still confused about clipping after reading all the articles.

You can prevent amplifier clipping by placing a limiter (a type of compressor) before the amplifier.  Some powered loudspeakers have this in their circuitry.  Regarding your question, an integrated amp is a preamp and a power amp built in a single cabinet.  They face the same "problems" regarding clipping.  Many classical pieces start out softly and build to very loud climaxes.  It's part of the music and most listeners want to hear the wide variation in volume levels.  In a well put together system with a knowledgeable listen it won't cause clipping.  Clipping won't damage most loudspeakers unless it's sustained for a relatively long period of time.

Right you are, you completely misunderstand some concepts. A big one is thinking of an amplifier as some kind of sentient being thinking about what it should do.

What I don’t understand is why would an amp allow itself to consume more power than it could handle in the first place.

Its a circuit. It doesn't "allow itself" anything. Because there is no "self". Its a circuit. It never "consumes more power than it can handle in the first place". It can't, the fuse would blow long before that could happen.  

What happens instead when you turn the volume up the incoming signal is able to be handled by all the circuits just fine until it gets to the output stage. Tubes or transistors, gloss over some details, basically two things happen. To understand what they are it helps to know a little about amplifiers.

Basically, AC power comes in from the wall, gets converted to DC in the power supply stage, and stored in power supply caps. Right away you can see there's three things that can limit power: the incoming AC, the power supply stage (basically a rectifier that converts AC to DC) and the power supply caps.  

Eventually there's either output transistors, or tubes and transformers. Either way these impose their own power limitations on the signal as well.

Nowadays we have these standardized power ratings that are great for leveling the field but they do have the unintentional side effect of blurring what's really going on. What you measure can actually contribute to confusing people more than helping them. Such sadly is the case here.

Because music isn't a uniform sine wave, its all different frequencies and amplitudes, constantly changing. The power supply has way more than enough power stored to put out huge power for an instant. The tubes or output transformers can also put out huge power for an instant. But do it long enough and either the power supply caps will drain, or the output devices heat up and become inefficient (heat is energy, the power is going to heat instead of sound) or both. Power drops off. A lot.

This is what we call clipping. The point where the power supply or the output devices, either one or both, simply cannot do any more. They are either too hot, or too drained. (Usually too hot.)

Same thing happens inside the speaker. Power goes into the voice coils, a very thin wire inside a magnetic field. The more power the hotter the voice coil gets. Eventually if it gets too hot resistance increases, heating accelerates, and if this goes on long enough the insulation starts to melt and typically the coil deforms and scrapes or even melts and sort of welds or glues itself in the gap.  

Power that is clean spikes up very high but only briefly. Power that is clipping goes very high and stays there. Worse, if you look real close at the flat clipped parts its not really flat. Its jumping up and down at a very high frequency. Its this high power high frequency part of clipping that blows tweeters, especially if there's a lot of high frequency in the music too.

This all happens the same with active or separate components. The only difference is if the speakers are active they can be designed to not fail no matter what signal goes in. Separate components offer a lot more performance but call for a little basic user knowledge. Which you now have.

@bwang29, what is the issue with active powered speakers that you are referring to ... a specific example ... ?

" 3. Another related confusion is, how is it possible that sometimes I see powered active speakers blown because it’s trying to play too loud? Would it be true that the amp in those active speaker should always be designed to operate within the limit of its power handling?"

Small playback systems are not intended to produce high SPL levels.
This reference is more about an issue of misuse.