Find the volume of the solid obtained by revolving about the y-axis the region bounded by `y=2-(x^2)/(2), y=0, x=1, and x=2`

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Using the method of cylindrical shells, the volume of the solid obtained by rotating about the y-axis the region bounded by `y=2-x^2/2, y=0, x=1` and `x=2` is given by:

`V=` `int_a^b2pixf(x)dx```

`rArr V=` `int_1^2 2pix(2-x^2/2)dx`

`rArr V=` `int_1^2 2pi(2x-x^3/2)dx`

`rArr V=2pi[x^2-x^4/8]_1^2`

`rArr V=2pi*9/8`

`rArr V=9/4pi`

Therefore, the volume of the required solid is `9/4pi` cubic units.

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