The solid that would be constructed by the curve y=(x^2) +1, the x-axis, y-axis and the line x=2 when rotated completely about x-axis, can be considered to be a series of cylinders of a height dx and a radius given by the value of y = x^2 + 1.

The volume would be:

Int [ pi*y^2 dx ], x = 0 to x = 2

=> Int [ pi*(x^2 + 1)^2 dx], x = 0 to x = 2

=> Int [pi* (x^4 + 2x^2 + 1) dx], x = 0 to x = 2

=> [pi* ( x^5/5 + (2/3)x^3 + x)], x = 0 to x = 2

=> pi*( 2^5/5 + (2/3)2^3 + 2 - 0)

=> 206*pi/15

**The required volume is 206*pi/15**

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