tape out = preout ???

If you plan on using the preamp's volume control, then using the tape output to feed the sub will not work because you are tying to match a variable output with a fixed output. I doubt that you will want to get up and adjust the sub level each time you adjust the volume control. Get a Y-splitter and split the preamp output, one for the mains and one for the sub.

Plato's right on. The tape out jacks have no volume control because they have a fixed output. The preouts should work? What kind of unit is it?

What Plato said. But a little more specifically, the tape out is going to pass the signal from the source without amplification or attenuation. When you hit a loud section from the source when connected thru the tape out you could damage your speakers, etc.

What I would consider doing is to get a pair of Y splitters and connect them to the pre-out, and have one set of connectors to the amp section and the other connected to the sub. That way the VC on the Integrated would control both the amp section on the Integrated and the sub.

I'm not familiar with your Integrated, this is just generic advise so take it for what its worth.

Looking at the manual for your integrated it specifically shows a diagram using the pre-outs to drive a sub-woofer while driving the main speakers from the speaker outputs (not pre-outs). If that does not work I would contact your dealer or Cambridge Audio. If the hookup is correct (it probably is) then there may be something wrong with the unit. What you are trying to do should work according the manual.

I see that the cambridge audio 840a v2 is an intergrated amp. therefore, the mains get hooked up to the speaker terminals and the sub get hooked up to the preouts - this is assuming that your sub is an active sub and not passive.