Are spiders part of the study of entomology?
Spiders are part of a taxon called Arachnida. Arachnida also contain scorpions, Oppiliones, acari, … The science of arachnids is logically called Arachnology

However, (wikipedia):


  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.

You will find here on tolweb.org more information about the systematics of Arthropoda