Find the exact area of the surface obtained by rotating the curve about the x-axis. y = 1 + ex , 0 ≤ x ≤ 9

The surface area is given by

since . To compute the integral, first let

so that , and the integral becomes

Next, let

so that . Then

so the integral becomes

Rewrite the integrand with

so that integrating the first term boils down to

where we substitute . Since

the first term in this integral contributes

The second term of the integral contributes

The antiderivative of is well-known (enough that I won't derive it here myself):

so this latter integral's contribution is

Then the surface area is

your answer is b ^w^ hope i