The definition of entomology says it is the science of studying insects.
I know spiders are not insects but does entomology include studying spiders as well?
Otherwise is there any name to the science of studying spiders (class Arachnida)?
In other words, in the informal sense entomology include arachnology but the strict definition entomology does not encompass arachnology.
As a general rule no. Spiders and other arachnids generally tend to have their own field of study called arachnology.