To find the surface area of the cone, we need to find the equation of the cone and the equation of the cylinder.

Given: z^2 = x^2 + y^2

To find the equation of the cylinder, we set x^2 + y^2 = 2x to get x^2 - 2x + y^2 = 0. Completing the square, we have (x - 1)^2 + y^2 = 1.

So the equation of the cylinder is (x - 1)^2 + y^2 = 1.

Now, to find the surface area of the cone inside the cylinder, we need to find the surface area of the cone and subtract the surface area of the part of the cone outside the cylinder.

The surface area of the cone is given by the formula A = πrℓ, where r is the radius of the base and ℓ is the slant height.

In this case, since z^2 = x^2 + y^2, we have z = √(x^2 + y^2).

The radius of the base, r, is the distance from the origin (0,0,0) to the point (x,y,z) on the cone, which is given by r = √(x^2 + y^2 + z^2) = √(x^2 + y^2 + x^2 + y^2) = √(2x^2 + 2y^2) = √2√(x^2 + y^2) = √2z.

The slant height, ℓ, is the distance from the point (x,y,z) on the cone to the point (x,y,0) on the base of the cone. This is given by ℓ = z.

So the surface area of the cone is A_cone = πrℓ = π(√2z)(z) = π√2z^2.

Now, to find the surface area of the part of the cone outside the cylinder, we need to find the points on the cone that lie outside the cylinder.

From the equation of the cylinder, (x - 1)^2 + y^2 = 1, we can see that the points (x,y,z) on the cone that lie outside the cylinder must satisfy (x - 1)^2 + y^2 > 1.

Substituting z^2 = x^2 + y^2, we have (x - 1)^2 + (z^2 - x^2) > 1.

Simplifying, we get z^2 - 2x + 1 > 1.

So z^2 - 2x > 0.

Since z^2 = x^2 + y^2, we have x^2 + y^2 - 2x > 0.

Completing the square, we have (x - 1)^2 + y^2 - 1 > 0.

So the points (x,y,z) on the cone that lie outside the cylinder satisfy the inequality (x - 1)^2 + y^2 - 1 > 0.

Therefore, the surface area of the part of the cone outside the cylinder is given by A_outside = ∫∫√2z