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If you read at digital video cameras specifications, you will probably read that camera has 20x optical zoom or even greater.
If you compare this camera with my Nikor 18-200 lens, the lens itself is much bigger then digital video camera, but digital video camera has bigger optical zoom. How this is made, and why my lenses are so big in compare with digital video cameras?
Term digital video camera is referencing to devices which is used for recording video in first place.
Your facts mislead you. Lens magnification and size have nothing to do with video or stills. They have to do with sensor-size.
Your DSLR has a large sensor and requires a large lens. The digital video camera you refer to has a small sensor and requires a small lens. You can very easily pick some random examples to show the opposite! Take a Canon Powershot SX50 HS, for example, which has a 50X optical zoom and a Sony NEX FS100U which has a sensor larger than your Nikon and will require an even larger lens to achieve the same magnification. This makes it impossible to get the kinds of zoom seen on small cameras without the lens requiring its own transport mechanism!
There are a few things that can be going on here. First, the wider the lens can get, the less telephoto it has to be able to go before being a high multiplier. Say your lens was able to do 10mm instead of 20, it was a pretty small change on the short end, but would be twice as large of a x multiplier.
Multipliers are not used in professional lens measurements for exactly that reason, it doesn't say much of anything about the actual extent of the range that can be covered.
Second, the size of the sensor is critical to the size of the lens. If you have a very small sensor, it is easy to use smaller optics since you are resolving to a smaller circle. You see a little bit of this when you see the effect that APS-C camera's have on the MM equivalent of lenses designed for full frame cameras.
So the combination of these factors results in a little point and shoot being able to have a significantly high multiplier for the zoom, but really it is shooting very wide and on a very small sensor (which also means a very large depth of field and very little if any background blurring.) A larger camera, like a DSLR, on the other hand has a much bigger sensor and typically doesn't go from ultra wide to super telephoto in a single lens, so the multipliers are smaller.
One final point as to why we don't see lenses for a very wide zoom range on DSLRs very often is that zoom lenses are a trade off. The wider the amount of zoom, the more compromises have to be made in the image quality of the optics. Prime (non-zoom) lenses almost always have the best image quality due to their simpler design and then short zoom ranges generally (though not always) have better image quality than their wider range of zoom counterparts (at least within a similar price range).
