For all parts of this problem, show all of your steps and cite any major theorems...

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For all parts of this problem, show all of your steps and cite any major theorems that you use. For each part, the answer should be an exact value. Let r(t) and y(t) be functions such that (r(t), y(t)), with 0 ≤ t ≤ 1, parametrizes a simple closed curve C₁ oriented counterclockwise in R². Also assume that (3r, -y)-nds=4. Part A) Use the flux form of Green's Theorem to compute the area of the region D bounded by C₁ in R². Part B) Now think of C₁ as lying in the ry-plane in R³, i.e. it is the curve parametrized by (r(t), y(t).0), with 0≤ t ≤ 1. Let F(x, y, z) = (y, 6x. 24), and compute the vector line integral $ F. dr Part C) Let x(t), y(t), and F be the same as in the previous parts. Now in addition let C₂ be the simple closed curve in R³ parametrized by (r(t), y(t), cos(8t) + 10), with 0 ≤ t ≤ 1. Compute fo F. dr For all parts of this problem, show all of your steps and cite any major theorems that you use. For each part, the answer should be an exact value. Let r(t) and y(t) be functions such that (r(t), y(t)), with 0 ≤ t ≤ 1, parametrizes a simple closed curve C₁ oriented counterclockwise in R². Also assume that (3r, -y)-nds=4. Part A) Use the flux form of Green's Theorem to compute the area of the region D bounded by C₁ in R². Part B) Now think of C₁ as lying in the ry-plane in R³, i.e. it is the curve parametrized by (r(t), y(t).0), with 0≤ t ≤ 1. Let F(x, y, z) = (y, 6x. 24), and compute the vector line integral $ F. dr Part C) Let x(t), y(t), and F be the same as in the previous parts. Now in addition let C₂ be the simple closed curve in R³ parametrized by (r(t), y(t), cos(8t) + 10), with 0 ≤ t ≤ 1. Compute fo F. dr