Standard chemotherapy for newly diagnosed epithelial ovarian, primary peritoneal, or fallopian tube cancer is usually a combination of 2 chemotherapy drugs called carboplatin and paclitaxel. This treatment has been found to be effective, but the cancer often comes back, requiring additional treatment. For this study, researchers will give carboplatin, bevacizumab, and weekly paclitaxel to see if the treatment is effective and safe in treating the disease.