Question 1
Find the arc length of the curve x =
![](data:image/png;base64, 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)
ln (1 + t
2), y = tan
-1 t, from t = 0 to t = 1.
◦ ln (
![](data:image/png;base64, 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)
- 1) units
◦ ln (
![](data:image/png;base64, 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)
+ 1) units
◦ ln (
![](data:image/png;base64, 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)
) units
◦ ln (3 -
![](data:image/png;base64, /9j/4AAQSkZJRgABAQAAAQABAAD/2wBDAAMCAgMCAgMDAwMEAwMEBQgFBQQEBQoHBwYIDAoMDAsKCwsNDhIQDQ4RDgsLEBYQERMUFRUVDA8XGBYUGBIUFRT/2wBDAQMEBAUEBQkFBQkUDQsNFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBT/wAARCAATABQDASIAAhEBAxEB/8QAHwAAAQUBAQEBAQEAAAAAAAAAAAECAwQFBgcICQoL/8QAtRAAAgEDAwIEAwUFBAQAAAF9AQIDAAQRBRIhMUEGE1FhByJxFDKBkaEII0KxwRVS0fAkM2JyggkKFhcYGRolJicoKSo0NTY3ODk6Q0RFRkdISUpTVFVWV1hZWmNkZWZnaGlqc3R1dnd4eXqDhIWGh4iJipKTlJWWl5iZmqKjpKWmp6ipqrKztLW2t7i5usLDxMXGx8jJytLT1NXW19jZ2uHi4+Tl5ufo6erx8vP09fb3+Pn6/8QAHwEAAwEBAQEBAQEBAQAAAAAAAAECAwQFBgcICQoL/8QAtREAAgECBAQDBAcFBAQAAQJ3AAECAxEEBSExBhJBUQdhcRMiMoEIFEKRobHBCSMzUvAVYnLRChYkNOEl8RcYGRomJygpKjU2Nzg5OkNERUZHSElKU1RVVldYWVpjZGVmZ2hpanN0dXZ3eHl6goOEhYaHiImKkpOUlZaXmJmaoqOkpaanqKmqsrO0tba3uLm6wsPExcbHyMnK0tPU1dbX2Nna4uPk5ebn6Onq8vP09fb3+Pn6/9oADAMBAAIRAxEAPwD9U6+StSe112Hxn8Qdc0v4peJbafxFqlsuneDPFd/YwaLZabKdNY+VHqFskvmSWM10VijaUG5K7X2B29A8SeLr79mO21Lxv8RfFtxrfgS7hgOtakbdgui6gZGQTQ2qeY4tJhJbWwhjLtC0ETuJ2uLq5TB0uw8feCfgT8OfBD/DzxFrd5daDCPF99oN/pYuVvXiVr1S9xeQ75553neS4UuSWdwxd96p3fw7+e3z/r01Lgk37239befY988Hy6bceEtEl0a6mvtHexgayurm4luJZoDGvlu8spMkjFcEu5LMTliSSaKl8NyeboGnN/ZE2gD7OgGlT+TvtBgARHyXePKjj5GZeOCRRVySUmkZRvyrm3J9W0mx1/Sr3TNTsrfUdNvYXtrqzu4llhnidSrxujAhlZSQVIIIJBq3RRUlBRRRQB//2Q==)
) units
◦ ln (2 -
![](data:image/png;base64, /9j/4AAQSkZJRgABAQAAAQABAAD/2wBDAAMCAgMCAgMDAwMEAwMEBQgFBQQEBQoHBwYIDAoMDAsKCwsNDhIQDQ4RDgsLEBYQERMUFRUVDA8XGBYUGBIUFRT/2wBDAQMEBAUEBQkFBQkUDQsNFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBT/wAARCAATABQDASIAAhEBAxEB/8QAHwAAAQUBAQEBAQEAAAAAAAAAAAECAwQFBgcICQoL/8QAtRAAAgEDAwIEAwUFBAQAAAF9AQIDAAQRBRIhMUEGE1FhByJxFDKBkaEII0KxwRVS0fAkM2JyggkKFhcYGRolJicoKSo0NTY3ODk6Q0RFRkdISUpTVFVWV1hZWmNkZWZnaGlqc3R1dnd4eXqDhIWGh4iJipKTlJWWl5iZmqKjpKWmp6ipqrKztLW2t7i5usLDxMXGx8jJytLT1NXW19jZ2uHi4+Tl5ufo6erx8vP09fb3+Pn6/8QAHwEAAwEBAQEBAQEBAQAAAAAAAAECAwQFBgcICQoL/8QAtREAAgECBAQDBAcFBAQAAQJ3AAECAxEEBSExBhJBUQdhcRMiMoEIFEKRobHBCSMzUvAVYnLRChYkNOEl8RcYGRomJygpKjU2Nzg5OkNERUZHSElKU1RVVldYWVpjZGVmZ2hpanN0dXZ3eHl6goOEhYaHiImKkpOUlZaXmJmaoqOkpaanqKmqsrO0tba3uLm6wsPExcbHyMnK0tPU1dbX2Nna4uPk5ebn6Onq8vP09fb3+Pn6/9oADAMBAAIRAxEAPwD9U6+StSe112Hxn8Qdc0v4peJbafxFqlsuneDPFd/YwaLZabKdNY+VHqFskvmSWM10VijaUG5K7X2B29A8SeLr79mO21Lxv8RfFtxrfgS7hgOtakbdgui6gZGQTQ2qeY4tJhJbWwhjLtC0ETuJ2uLq5TB0uw8feCfgT8OfBD/DzxFrd5daDCPF99oN/pYuVvXiVr1S9xeQ75553neS4UuSWdwxd96p3fw7+e3z/r01Lgk37239befY988Hy6bceEtEl0a6mvtHexgayurm4luJZoDGvlu8spMkjFcEu5LMTliSSaKl8NyeboGnN/ZE2gD7OgGlT+TvtBgARHyXePKjj5GZeOCRRVySUmkZRvyrm3J9W0mx1/Sr3TNTsrfUdNvYXtrqzu4llhnidSrxujAhlZSQVIIIJBq3RRUlBRRRQB//2Q==)
) units
Question 2
Find the length of one arch of the cycloid x = a(θ - sinθ), y = a(1 - cosθ).
◦ 8a units
◦ 12a units
◦ 4a units
◦ 10a units
◦ 6a units