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)?
Entomology (from Greek ἔντομος, entomos, "that which is cut in pieces or engraved/segmented", hence "insect"; and -λογία, -logia) is the scientific study of insects, a branch of arthropodology. In the past the term "insect" was more vague, and historically the definition of entomology included the study of terrestrial animals in other arthropod groups or other phyla, such as arachnids, myriapods, earthworms, land snails, and slugs. This wider meaning may still be encountered in informal use.
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.
The term entomology is derived from the Greek words Entomo ("Insect") and Logia(word) . Considering that Entomo generally refers to insect(s) the question is whether Arachnids could be considered entoma in Greek as there is no other used term that refers to arachnids.