Finding volume by rotation of a triangleThe following problem is from the book, Calculus and Analytical Geometer by Thomas and Finney. Problem:Find the volume generated by revolving the following region: The triangle with vertices (1,1), (1,2) and (2,2) about the y-axis.Answer:Since we are going around the y-axis, our integral will integrate with respect to y not x. Observe that the points (1,1) and (2,2) are on the line y = x. Let v be the volume we seek. v = π ∫ 1 2 2 2 − y 2 d y = π ( 4 y − y 3 3 ) | 1 2 v = π ( 8 − 8 3 − ( 4 − 1 3 ) ) = π ( 8 − 8 3 − 4 + 1 3 ) v = π ( 4 − 7 3 ) v = 5 π 3 The book's answer is 4 π 3 . What did I do wrong?

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Finding volume by rotation of a triangleThe following problem is from the book, Calculus and Analytical Geometer by Thomas and Finney. Problem:Find the volume generated by revolving the following region: The triangle with vertices (1,1), (1,2) and (2,2) about the y-axis.Answer:Since we are going around the y-axis, our integral will integrate with respect to y not x. Observe that the points (1,1) and (2,2) are on the line   y  =  x. Let v be the volume we seek.                    v                            =        π                  ∫          1          2                          2          2                −                  y          2                                d        y        =        π                  (          4          y          −                                    y              3                        3                    )                                                    |                                1          2                                    v                            =        π                  (          8          −                      8            3                    −                      (            4            −                          1              3                        )                    )                =        π                  (          8          −                      8            3                    −          4          +                      1            3                    )                                    v                            =        π        (        4        −                  7          3                )                            v                            =                              5            π                    3                    The book&#039;s answer is             4      π        3  . What did I do wrong?

Kendrick Mendez · Accepted Answer

Explanation:When the triangle is rotated, a &quot;washer&quot; of outer radius y and inner radius 1 is formed. Thus, the volume should be given by   π      ∫          1              2            y    2    −  1  d  y, which should evaluate to   4  π      /    3.

Jaydan Ball · Accepted Answer

Explanation:  V  =  π      ∫    1    2    (      y    2    −  1  )  d  y  =  π      (                  y        3            3        −    y    )                      |              1    2    =  π      (          8      3        −    2    −          (              1        3            −      1      )        )    =  π      (          2      3        +          2      3        )    =            4      π        3    .I think, the mistake is that you rotated around x-axis.

Find the volume generated by revolving the following region: The triangle with vertices (1,1), (1,2) and (2,2) about the y-axis.

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