MAT 237 Y
SAMPLES OF PREVIOUS YEAR'S TESTS
TEST #3. COURSE 2003-2004.
1. (15 marks) Evaluate the line integral where C is the curve of intersection of the surfaces
2 x + y = 5 and x 2 + z = 7 from the point ( 0 , 5 , 7 ) to the point ( 2 , 1 , 3 ) .
2. (15 marks) Consider the surface S in R 3 parametrized by F ( u , v ) = ( u cos v , u sin v , 2 ) with
0 £ u £ 1 and 0 £ v £ p . Compute .
3. (15 marks) Evaluate the improper integral , where W is the
solid region consisting of all the points ( x , y , z ) which are inside the cone
but outside the sphere x 2 + y 2 + z 2 = 3 .
4. (20 marks) Evaluate the double integral , where R is the region in the first quadrant,
enclosed between the curves x y = 1 and x y = 9 and between the lines x = y and x = 16 y .
Hint: Make the change of variables x = u v , y = u / v with u > 0 and v > 0 .
5. (20 marks) Let R be the solid region defined by the conditions 0 £ z £ 2 – and x 2 + y 2 £ 2 y
and let d ( x , y , z ) = be its mass density at any point ( x , y , z ) . Find the total mass of the solid R .
6. (15 marks) Let S be the surface a x 2 + a y 2 + z 2 = a , where a denotes a positive real number.
Suppose that g ( a ) denotes the flux of the vector field F ( x , y , z ) = x i + y j + a z k out of the closed surface S .
Find the value of a , if any, for which g ( a ) = 6 a p .