Math 161

Spring 2000

Final Exam

 

  1. (20 pt) Evaluate the following integrals:

a)     b)            c)        d)

 

  1. (20 pt) Consider the region bounded by the function  and the

a)      Find the volume obtained when this curve is revolved about the

b)      Find the volume obtained when this curve is revolved about the

c)      Find the surface area obtained when this curve is revolved about the

d)      Find the centroid of the region.

 

  1. (12 pt) Determine if the following sequences converge or diverge.

a)     b)        c)

 

  1. (15 pt) Determine if the following series converge or diverge.

a)          b)              c)

 

  1. (10 pt) Consider the polar equation .

a)      Sketch the graph and show your work.

b)      Find the area enclosed by the inner loop of this figure.

 

  1. (12 pt) Consider the function

a)      Find a Maclaurin series for

b)      Use your Maclaurin series to show that  

c)      Find  with error less than or equal to 1/10,000,000.

  1. (10 pt) Graph the curve traced out by the parametric equations  Indicate where the tangent line is horizontal and where the tangent line is vertical.
  2. (5 pt) Find the area enclosed by the ellipse  with (Hint: Consider the parametric equations
  3. (6 pt) Find the interval of convergence of the power series .

 

 

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