compute surface areas by using surface integrals for the following surfaces: (i) z = x^2 + y^2, z = - (x^2 + y^2)

To compute the surface area using surface integrals, we first need to parameterize the surface.

(i) For the surface z = x^2 + y^2, we can use cylindrical coordinates (r, θ, z) where x = rcos(θ), y = rsin(θ), and z = r^2. The differential area element in cylindrical coordinates is dS = rdθdz.

The surface integral for the area is then given by:

∫∫_S dS = ∫ (from 0 to 2π) ∫ (from 0 to R) r dr dθ

This integral evaluates to πR^4, where R is the radius of the cylinder.

(ii) For the surface z = -(x^2 + y^2), we can again use cylindrical coordinates. The differential area element is the same, dS = rdθdz.

The surface integral for the area is then given by:

∫∫_S dS = ∫ (from 0 to 2π) ∫ (from -R to 0) -r dr dθ

This integral also evaluates to πR^4.

So, the surface areas for both surfaces are the same and equal to πR^4.