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Yes. In fact, they are integers.
Answer: 1, 64, & 729
The cube root of 5000 is approx 17.1 So the numbers 1 to 17 have cubes which are smaller than 5000 that is, there are 17 such numbers.
If you include fractions and decimals, then there are an infinite number of squares and cubes in any range. If you consider only whole numbers, then from 1 to 20: Squares: 1, 4, 9, and 16. Cubes: 1 and 8.
Usually they don't, unless you work in engineering.