Not sure about intergers but integers are regularly used.
The scale of most maps is defined in terms of an integer value: 1 cm = 10 kilometres or somehing like that.
Latitudes and longitudes are often given as integers. At a more detailed level, Northings and Eastings are also marked on maps as integers.
Physical maps may give contours and these are usually given for selected integer values. Other physical attributes - rainfall, temperatures (isotherms) etc also use integers.
In recent years technology has enabled mapping of other socio-economic data. For example, some cities have produced mappings of crimes classified to areas. At a high level, these would be the [integer] number of crimes. There are many such possibilities.
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Every aspect of mathematics depends upon integers, and even if you don't do any mathematics, you count things with integers. If you even want to know how many guests you are having for dinner, you need to use integers.
You can sometime use integers but often speed or velocity can also be irrational.
Yep.
Yes.
You can use them for counting, which is in fact addition. You can also use them for subtracion. Integers are natural numbers. This means you don't use decimals or fractions.