`y = 2 - (1/2)x, y = 0, x = 1, x = 2` Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a...

`y = 2 - (1/2)x, y = 0, x = 1, x = 2` Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer. (about the x-axis)

Expert Answers
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The volume of the solid obtained by rotating the region bounded by the curves `y=2 - x/2, y=0, x=1,x=2` , about x axis, can be evaluated using the washer method, such that:

`V = int_a^b pi*(f^2(x) - g^2(x))dx`

Since the problem provides you the endpoints x=1,x=2, you may find the volume such that:

`V = int_1^2 pi*(2 - x/2 - 0)^2 dx`

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

`V = pi*int_1^2 (4 - 2x + x^2/4)dx `

`V = pi*(int_1^2 dx - 2int_1^2 x dx + (1/4)int_1^2 x^2 dx)`

`V = pi*(x - x^2 + x^3/12)|_1^2`

`V = pi*(-2 + 2^2 - 2^3/12 - 1 + 1^2 + 1^3/12)`

`V = pi*(2 - 8/12 + 1/12)`

`V = (17pi)/12`

Hence, evaluating the volume of the solid obtained by rotating the region bounded by the curves `y=2 - x/2, y=0, x=1,x=2` , about x axis , using the washer method, yields `V = (17pi)/12.`