In this question we use two ways to calculate an integral and in doing so, verify the Generalized Stokes's Theorem. Integrate over the oriented boundary of the manifold Min R³ consisting of those points x satisfying Part 1. First we compute = ydx / dz, ²+590≤3. 4. Since M is a cylinder in R³, it's boundary consists of three parts, and we integrate each one separately. вм For the side surface S₁ of the cylinder, we parametrize it using = 3 cos, y = 3 sinu and z = z. So P₁ = (C ◉◉ [+] (2) [L <= dudz= For the disc S₂ at the top of the cylinder, we parametrize it using dudz= P₁ = Pr оо Op Op 4= $ B₂ Dr Ou Finally for the disc Sy at the bottom of the cylinder, we parametrize it using drdu= P₁ = (0 = др др drdu= Dr' Ou +2 Part 2. Now let's compute q= q= 4== -L- Now: M M оо do. To begin we parametrize M: O O O ㅁㅁㅁ dsdud:= drdu= p = (3 cosu, 3 sinu, z), u € [0,2], = = [0,3]. p = (rcosu,rsinu, 3), [0, 2π], r € [0,3]. p= 14 p=(rcosu, rsinu, 0), [0,2], † € [0,3], q= (scosu, 8sinu, ), 8€ [0,3], µ = [0,2], z ¤ [0,3].