Question

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

Answer 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