Which of the following is the volume of the solid obtained b

Which of the following is the volume of the solid obtained by revolving the region bounded by y = x2 and y = 4x - x2 about the line x = 2. 8 pi 16 pi /3 8 pi /3 8 pi /3 32 pi/3

Solution

volume of outer solid :   (2-x)² dy :   (2-x)² (4 -2x)dx

: 2(2 -x)³ dx

: (-/2) (2 - x)⁴   from x =0 to x=2

: (-/2)[ 0 - 16 ]

: 8  

volume of inner solid :

(2 -x)² dy

: (2-x)² 2x dx

: 2 [ x⁴/4 + 2x² - 4x³ /3 ] from x=0 to x=2

: 2 [ 4 + 8 - 32/3 ]

: 2 [ 4/3 ]

: 8/3

volume of desired : 8 - 8/3

: 16/3 answer !!!

b) is the correct option !!!