Find the exact area of the surface z = 1 + 2x + 3y + 4y2, 1 ≤ x ≤ 3, 0 ≤ y ≤ 1.

Parameterize the region by

with and. Then the area is given by the surface integral

You have

and the cross product is

with norm.

So the integral reduces to

The integrand is independent of, so you have

Let, so that. Then the integral is equivalent to

I'll leave the rest as an exercise for you, but you should end up with a result resembling