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					           AREAS & VOLUMES OF REVOLUTION
                          Area enclosed by y = 1 – x2 and the x- & y-axes
                                                       1
                          Area =  y dx             = 0 1 – x2 dx
                          Volume of revolution about x–axis
                                                         1
                                  =  y2 dx        = 0 (1 – x2)2 dx
                          Volume of revolution about y–axis
                                                         1
                                  =  x2 dy        = 0 1 – y dy
                          Area enclosed by y = 1–x2 & the lines x = 1, y =1
                                                       1
                          Area =  1 – y dx         = 0 1 – (1 – x2) dx
                          Volume of revolution about x–axis
                                                                 1
                                  = r2h –  y2dx = 121 – 0 (1 – x2)2 dx
                          Volume of revolution about y–axis
                                                                    1
                                  = r2h –  x2 dy = 121 – 0 1 – y dy
                          Area enclosed by f(x) = x2 + 1 & g(x) = 2x2
f(x) = x2 + 1                                          1
                          Area =  f(x) – g(x) dx = 0 (x2 + 1) – 2x2 dx
                          Volume of revolution about x–axis
                                  =  [f(x)]2 dx –  [g(x)]2 dx
                                       1                   1
                                  = 0 (x2 + 1)2 dx – 0 (2x2)2 dx
              g(x) = 2x2
                          Volume of revolution about y–axis
                                  =  (g–1(y))2 dy –  (f–1(y))2 dy
                                       2 y         2
                                  = 0 dy – 1 y – 1 dy
                                         2
                          Area enclosed by f(x) = 2 – x2 & g(x) = x2
            f(x) = 2 – x2                            1
                          Area =  f(x) – g(x) dx = 0 (2 – x2) – x2 dx
                          Volume of revolution about x–axis
                                  =  [f(x)]2 dx –  [g(x)]2 dx
                                       1
                                  = 0 (2 – x2)2 – (x2)2 dx
                g(x) = x2 Volume of revolution about y–axis
                                  =  [g–1(y)]2 dy +  [f–1(y)]2 dy
                                       1           2
                                  = 0 y dy + 1 2 – y dy
                          Area enclosed by y = ex and the lines x = 1, y = 1
                                                       1
                          Area =  y – 1 dx         = 0 ex – 1 dx
                          Volume of revolution about x–axis
                                                          1
                                  =  y2 dx – r2h = 0 (ex)2dx – 12 1
                          Volume of revolution about the line y = 1
                                                        1
                                  =  (y – 1)2 dx = 0 (ex – 1)2 dx
                          Volume of revolution about the line x = 1
                                       2
                                  = 1 (ln y – 1)2 dy i.e. rotate y = ex+1

				
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