Paclitaxel and Carboplatin With or Without Bevacizumab in Treating Patients With Stage II, Stage III, or Stage IV Ovarian Epithelial Cancer, Primary Peritoneal Cancer, or Fallopian Tube Cancer

Inclusion Criteria: * Primary Surgery and Neoadjuvant Chemotherapy with Interval Cytoreductive Surgery Patients: * Patients must have measurable disease; at least one target lesion must have a minimum length of 1 cm in both the long and short axis (determined at the local site); for primary surgery patients, if no radiographic evidence of measurable disease is obtained prior to registration this can be based on surgical findings; imaging then would need to be completed in the 14 days between Gynecology Oncology Group (GOG) registration and chemotherapy initiation * After GOG registration, the American College of Radiology \[ACR\] Imaging Core Laboratory will confirm target lesion as required per protocol; the GOG-eligibility (RECIST) scan and baseline T0 perfusion CT scans will be reviewed prior to the intermediate T1 perfusion CT time point * Primary Surgery Patients: * Patients with a histologic diagnosis of epithelial ovarian cancer, peritoneal primary carcinoma or fallopian tube cancer, stage II -IV suboptimally debulked (any residual disease \> 1 cm); International Federation of Gynecology and Obstetrics (FIGO) stage is assessed following the completion of initial abdominal surgery, appropriate imaging studies and with appropriate tissue available for histologic evaluation; the minimum surgery required is an abdominal surgery providing tissue for histologic evaluation and establishing and documenting the primary site and stage; if additional surgery was performed, it…