Calculus HW

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1. (2 points) Consider ∫ 𝑓(𝑥,𝑦)𝑑𝐴 = 𝑅

∫ ∫ (5𝑦 + 6) √4−𝑥2

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)

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3. (4 points) Let 𝑊 be the solid bounded by the following two surfaces:

Surface 𝑆1: 𝑧 = √6 − 𝑥 2 − 𝑦2

Surface 𝑆2: 𝑧 = 𝑥 2 + 𝑦2

(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)