Use Pappus's Theorem to find the volume of the solid of revolution obtained by rotating the triangular plane region specified by 0 ≤ y ≤ 1 -
![](data:image/png;base64, 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)
about (a) the line x = 2 and (b) the line y = 2.
◦ (a) 2π units
3, (b)
![](data:image/png;base64, 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)
units
3◦ (a) 4π units
3, (b)
![](data:image/png;base64, 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)
units
3◦ (a) 4π units
3, (b)
![](data:image/png;base64, /9j/4AAQSkZJRgABAQAAAQABAAD/2wBDAAMCAgMCAgMDAwMEAwMEBQgFBQQEBQoHBwYIDAoMDAsKCwsNDhIQDQ4RDgsLEBYQERMUFRUVDA8XGBYUGBIUFRT/2wBDAQMEBAUEBQkFBQkUDQsNFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBT/wAARCAAmABwDASIAAhEBAxEB/8QAHwAAAQUBAQEBAQEAAAAAAAAAAAECAwQFBgcICQoL/8QAtRAAAgEDAwIEAwUFBAQAAAF9AQIDAAQRBRIhMUEGE1FhByJxFDKBkaEII0KxwRVS0fAkM2JyggkKFhcYGRolJicoKSo0NTY3ODk6Q0RFRkdISUpTVFVWV1hZWmNkZWZnaGlqc3R1dnd4eXqDhIWGh4iJipKTlJWWl5iZmqKjpKWmp6ipqrKztLW2t7i5usLDxMXGx8jJytLT1NXW19jZ2uHi4+Tl5ufo6erx8vP09fb3+Pn6/8QAHwEAAwEBAQEBAQEBAQAAAAAAAAECAwQFBgcICQoL/8QAtREAAgECBAQDBAcFBAQAAQJ3AAECAxEEBSExBhJBUQdhcRMiMoEIFEKRobHBCSMzUvAVYnLRChYkNOEl8RcYGRomJygpKjU2Nzg5OkNERUZHSElKU1RVVldYWVpjZGVmZ2hpanN0dXZ3eHl6goOEhYaHiImKkpOUlZaXmJmaoqOkpaanqKmqsrO0tba3uLm6wsPExcbHyMnK0tPU1dbX2Nna4uPk5ebn6Onq8vP09fb3+Pn6/9oADAMBAAIRAxEAPwD9D/iT4+1vwZ4v+H9haWlhJpXiHW10m4mmZ2nXNpdzkoowq4+zIMktne3yjaCeH+Inxv8AFXhCHxT4vt10GLwF4U1aHTNStL+GUXtxGTAJ7qO5WXZEIzOcRtC5cRE703DHXfF/4ZeKfiBr/gi/0HxPpGgQeGtU/tcwajoct+1zN5E0AXel3AETZcScbWO7acgAq2ZqvwK1a/8AEerpF4siTwRrWrwa3qOhXGmGW6NxEYWMcN0JlEcLtBGWRonbl9rru4OzXnv30tfytzf8B6ly5W428r+fxXt5/D5b+hNp37UvgPVNRsLKGPxfHNfXUdlA1z4G1yCJpnUMiGSSzVFypDZYgBfmOBzXrdeU+Ivg7rl7F4bu9F8Xppmu6P4hutea5vdOa7tbsTpcRPA8AnRgFiuNqMsg27AcHJFerVTtbTf/AIb+v0XWOv8AXn/X6s8q/wCGTvgh/wBEb+H/AP4S9j/8arzfxp8Fvhv4Q8W6Hb/8M1/Di98MalqcGknUE02xF+kspwJVtBaFXhXksxmVwFdvLIXJ+na8O+JPwW1P4r+OLK41nw14RtIdLvLe60zxvazyNr9rHFOk3kRRm3Hlb9roXW5Iw5PlHJWs7vmj+Pp1B/C+/T16G5/wyd8EP+iN/D//AMJex/8AjVelaTpNjoGlWWmaZZW+nabZQpbWtnaRLFDBEihUjRFACqqgAKAAAABVuiqAKKKKACiiigD/2Q==)
units
3◦ (a) 2π units
3, (b)
![](data:image/png;base64, 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)
units
3◦ (a) 2π units
3, (b)
![](data:image/png;base64, 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)
units
3