`int_0^2(piy)/(1+y^2)dy` This integral represents the volume of a solid. Describe & draw the solid.

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to remember how to evaluate the volume of a solid obtained by rotating a region, under a given curve, about x or y axis, using the method of cylindrical shells, such that:

`V = int_a^b 2pi*x*f(x)dx` (solid obtained by rotating the region under the curve `f(x)` , around y axis)

`V = int_a^b 2pi*y*f(y)dy` (solid obtained by rotating the region under the curve `f(y)` , around x axis)

Comparing these formulas to the integral provided by the problem, yields that the solid is obtained by rotating the region under the curve `f(y) = x = 1/(2(1 + y^2))` , around x axis.

You need to evaluate the curve` f(x)` , hence, you need to re-write the expression `x = 1/(2(1 + y^2))` such that y to be expressed in terms of x:

`x = 1/(4(1 + y^2)) => 4x + 4xy^2 = 1 => 4xy^2 = 1 - 4x`

`y^2 = (1 - 4x)/(2x) => y = sqrt((1 - 4x)/(2x))`

Hence, the given integral represents the formula of the volume of the solid obtained by rotating the region `0<= y <= sqrt((1 - 4x)/(2x))` , from 0 to 2, around x axis.

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