1. (2 points) Consider ∫ 𝑓(𝑥, 𝑦)𝑑𝐴 = 𝑅 ∫ ∫ (5𝑦 + 6) √4−𝑥 2 1 √3 −1 𝑑𝑦𝑑𝑥. (A) Sketch the region, 𝑅 of the integration above. (B) Rewrite the integral in the order of 𝑑𝑟𝑑𝜃 in polar coordinates. (solution) 2. (2 points) Set-up an integral and compute the integral to find the total mass of a thin semi-circular disc with radius of 4, whose mass density is proportional to the distance from the center of the disc. (solution) Page 3 of 3 3. (4 points) Let 𝑊 be the solid bounded by the following two surfaces: Surface 𝑆1: 𝑧 = √6 − 𝑥 2 − 𝑦 2 Surface 𝑆2: 𝑧 = 𝑥 2 + 𝑦 2 (A) Sketch the solid 𝑊. (solution) (B)-(C) Use the specified order of coordinates given below to express (NOT evaluate) the volume of the solid. [You WILL NOT separate integrals unless you have to.] (B) Cartesian coordinates in the order of 𝑑𝑦𝑑𝑥. (solution) (C) Polar coordinates in the order of 𝑑𝑟𝑑𝜃. (solution) (D) Cartesian coordinates in the order of 𝑑𝑧𝑑𝑦𝑑𝑥. (solution)