ENGI 2422 Engineering Mathematics 2

Faculty of Engineering and Applied Science
2008 Winter

Problem Set 11   Questions

1. Evaluate

   

   over the triangular region D that is bounded by the lines   y = x,   y = –x   and   x = 2 .

2. Evaluate

   

   over the region R that is bounded by the lines   y = 1 + x,   y = 1 – x   and   y = 0 .

3. Evaluate

   

   over the region R that is bounded by the triangle whose vertices are the points (0, 0), (2, 1) and (1, 2):

   1. directly
   2. using the transformation of variables   x = 2u + v,   y = u + 2v   [bonus].

4. Find the mass and the location of the centre of mass of the lamina D defined by
   { 0 < x < 2 , –1 < y < 1 } and whose surface density is   σ = xy² .

5. Find the location of the centre of mass of the lamina D defined by the part of   x² + y² < 1   that lies in the first quadrant and whose surface density is directly proportional to the distance from the x-axis.

6. Evaluate

   

   where R is the region in the first octant that is between 1 and 2 units away from the origin.

7. [Bonus.]
   Use the transformation of variables   x = u / v,   y = v   to evaluate

   

   over the region R (in the first quadrant) that is bounded by the lines   y = x/3, y = 3x   and the hyperbolae   xy = 1 and xy = 3 .

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