`y = sqrt(x), x = 0, y = 2` Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by...

The shell has the radius y, the cricumference is `2pi*y` and the height is `y^2 - 0` , hence, the volume can be evaluated, using the method of cylindrical shells, such that:

`V = 2pi*int_0^2 y*(y^2 - 0) dy`

`V = 2pi*int_0^2 y^3 dy`

Using the formula `int y^n dy = (y^(n+1))/(n+1) ` yields:

`V = 2pi*(y^4)/4|_0^2`

`V = pi*(y^4)/2|_0^2`

`V = pi*(2^4 - 0^4)/2`

`V = 8pi`

Hence, evaluating the volume, using the method...

The shell has the radius y, the cricumference is `2pi*y` and the height is `y^2 - 0` , hence, the volume can be evaluated, using the method of cylindrical shells, such that:

`V = 2pi*int_0^2 y*(y^2 - 0) dy`

`V = 2pi*int_0^2 y^3 dy`

Using the formula `int y^n dy = (y^(n+1))/(n+1) ` yields:

`V = 2pi*(y^4)/4|_0^2`

`V = pi*(y^4)/2|_0^2`

`V = pi*(2^4 - 0^4)/2`

`V = 8pi`

Hence, evaluating the volume, using the method of cylindrical shells, yields `V = 8pi.`