Given information: Given curve y=2−12x

Calculation:

The region is shown in the picture:

Rotate about x−axis , get polygon shown in the picture:

When we slice through the point x ,we get a disk with radius y=2−12x:

The area of this cross- section is:

  A(x)=π(2−12x)2        =π(4−2⋅2⋅12x+14x2)       =π(4−2x+14x2)

The solid lies between x=1 and x=2 ,so its volume is:

  V=∫12A(x)dx  =∫12π(4−2x+14x2)dx   =π∫12(4−2x+14x2)dx   =π((4x−2x22+14x33)]02  =π[(4⋅2−22+14233)−(4⋅1−12+14133)]  =π(8−4+23−(4−1+112))  =π(4+23−4+1−112)  =π(8+12−112)  =19π12

Thus, volume of the solid obtained by rotating the region bounded by the given curves is 1912π