How many 1's we get

05AB1E, 5 4 bytes

Thanks a lot to OP! Now I can save a byte

ŸSΘO

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Explanation

Ÿ    Inclusive range
 S   Split the string into individual chars
  Θ  (Vectorizes) Does this character == "1"?
   O Sum the resulting list

Python 2, ³⁹ 36 bytes

-3 bytes thanks to @SurculoseSputum

lambda a,b:`range(a,b+1)`.count('1')

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Python 3, ⁴² 40 bytes

-2 bytes thanks to @JoKing

lambda a,b:f"{*range(a,b),b}".count('1')

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Rust+itertools, 42 bytes

|a,b|(a..=b).join("").matches('1').count()

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