# Volume by nuhman10

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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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