ENGI 2422 Engineering Mathematics 2

Faculty of Engineering and Applied Science
2008 Winter

Problem Set 2   Questions

Arc Length, Curvature:

1. For the curve defined parametrically by   r(t) = (t³ + 2) i + (3t² + 1) j,
   1. Find the length along the curve from   t = 0   to   .
   2. Find the curvature   (t).

2. For the curve defined parametrically by   r(u) = (cos²u) i + (sin²u) j,
   1. Find the length along the curve from   u = 0   to   .
   2. Find the curvature   (u).
   3. Find the Cartesian equations (in ³) for this curve.

3. For the curve, whose equation is expressed in terms of the parameter   t   by
                 r(t)   =   3e^–t cos t i   +   3e^–t sin t j   +   4e^–t k ,
   1. Find the distance along the curve from the point (3, 0, 4) to the origin.
   2. Find the curvature   (t)   and the radius of curvature   (t).
   3. Describe the behaviour of the curve as the parameter  

Conic Sections and Quadric Surfaces:

4. Classify the following conic sections and identify the major axis, where applicable:

   1. 16 x² – 9 y² = 144
   2. 16 x² + 9 y² = 144
   3. 16 x – 9 y² = 0
   4. 16 x² – 9 y² = 0

5. Classify and sketch the following quadric surfaces:
   1. x² – 2 y² + 3 z² = 4
   2. x² – 2 y² + 3 z² = –4
   3. x² – 2 y² + 3 z² = 0
   4. x² + 2 y² + 3 z² = –4
   5. 16 x² + 9 y² = 144
   6. 16 z – 9 y² = 0

Surfaces of Revolution:

6. Write down the equation of the surface of revolution formed when the curve   y = f (x)   is rotated about the indicated line
   and
   evaluate the area of the curved surface generated by the revolution, between x = 0 and x = 1 :

   1. y = f (x) = 2 x   about the x-axis
   2. y = f (x) = 2 + x²   about the line   y = 1
      In part (b), do not attempt to evaluate your definite integral.

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