`x^2 - y^2 = 7, x = 4`Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curve about the specified...

The shell has the radius `5 - y` , the cricumference is `2pi*(5 - y)` and the height is `4 - x` , hence, the volume can be evaluated, using the method of cylindrical shells, such that:

`V = 2pi*int_(y_1)^(y_2) (5 - y)*(4 - x) dy`

`V = 2pi*int_(y_1)^(y_2) (5 - y)*(4 - sqrt(7+y^2)) dy`

You need to find the endpoints, using the equation `sqrt(7+y^2) = 4 => 7+y^2 = 16 => y^2=9 => y_1=-3, y_2=3`

`V = 2pi*int_(-3)^(3) (20 - 5sqrt(7+y^2) - 4y + y*sqrt(7+y^2)) dy`

`V = 2pi*(int_(-3)^(3) 20dy - 5int_(-3)^(3)sqrt(7+y^2) dy - 4int_(-3)^(3) ydy + int_(-3)^(3) y*sqrt(7+y^2) dy)`

`V = 2pi*(20y - (5/2)*sqrt(y^2+7) - (35/2)sinh^(-1) (y/sqrt7) - 2y + (2/3)sqrt((7+y^2)^3))|_(-3)^(3)`

`V = 2pi*(60 - 10- (35/2)sinh^(-1) (3/sqrt7) - 6 + (2/3)64 + 60 - 10 + (35/2)sinh^(-1) (-3/sqrt7) - 6 - 128/3)`

`V = 2pi*(88 - (35/2)sinh^(-1) (3/sqrt7) + (35/2)sinh^(-1) (-3/sqrt7))`

Hence, evaluating the volume, using the method of cylindrical shells, yields `V = 2pi*(88 - (35/2)sinh^(-1) (3/sqrt7) + (35/2)sinh^(-1) (-3/sqrt7)).`