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