Use the shell method to find the volume of the solid generated by revolving the region bounded by the line y=x+2 and the parabola y=x^2 about the following lines.
(a) the line x=2
(b) the line x=-1
(c) the x-axis
(d) the line y=4

8 π or

25.13 unit^3 to the nearest hundredth.

Step-by-step explanation:

(A)

The height of the shell is (2 + x - x^2) and  the radius is (2 - x).

V = 2π ∫(2 - x)(x + 2 - x^2) dx  between the limits x = 0 and x = 2.

=  2π ∫ (2x + 4 - 2x^2 - x^2 - 2x + x^3) dx

 = 2π ∫ (x^3 - 3x^2 + 4 ) dx

= 2π [ x^4/4 - x^3 + 4x ]  between x = 0 and x = 2

= 2 π [4 - 8 + 8 )

= 2 π * 4

= 8π

= 25.13 unit^3 to the nearest hundredth.