 # (1) Solve the following differential equation. (8 pts.) dy = (8r++...

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(1) Solve the following differential equation. (8 pts.) dy = (8r++3)e-34, = da (2) Find the area of the surface obtained by rotating the curve given by f3 x (t) =t T3, y(t) = t2. , 0 < t < 2 about the x-axis. (8 pts.) (3) A bacteria culture doubles in size every 5 hours, how long will the culture take to triple in size? (8 pts.) (4) Find the length of the curve given by r = e20, , O(5) Find the centroid of the region bounded by the curves y = 3 - x2 and y = - -1. (8 pts.) (6) Consider the parametric equations x = -2 COS 0, y = sin² 0, , 0 < 0 < TT. (a) Sketch the curve represented by the parametric equations. (2 pts.) (b) Describe the motion of a particle with position (x, y) as 0 varies in the interval 0, \T]. (2 pts.) (c) Find the equation of the tangent line to the curve at 0 = 37/4. (2 pts.) d2y (d) Find the at 0 = 37/4. (2 pts.) d.c² (7) (a) Graph r = 2 + COS 0 and r = 2 on the same polar axis. (3 pts.) (b) Find the area inside the curve r = 2 and outside the curve r = 2 + cos 0. (5 pts.) Note cos2 0 = 1/1 (1 + COS 20). (8) Find polar or Cartesian coordinates for the given points. (2 pts. each) (a) (x, y) = (-2,2) (r,0) = (b) (x,y) = (1, - V 3) (r,0) = (c) (r,0) = (2,7x/6) (x,y) = (d) (r,0) = = (212,57/4) (x,y) =

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