MULTIVARIABLE CALCULUS

COURSE 2005-2006

SAMPLES OF PREVIOUS YEAR'S TESTS

TEST #3. COURSE 2004-2005.

1. a) (10 marks) Evaluate the path integral of the function  f ( x , y , z ) = x + y – 4 z  along the path

c : t ® ( 2 t – 2 sin t , 2 t – 2 cos t , t ) ,  0 ≤ t ≤ π / 4 .

b) (16 marks) Evaluate the triple integral   , where  W  is the solid region bounded by the planes

y = x ,  y = 2 x ,  y = 2 – 2 z  and  z = 0 .

2. a) (10 marks) Let  S  be the surface parametrized by  Ф ( u , v ) = ( u + v , u – v , u ² + v ² ) , where

u ² + v ² ≤ 3 . Compute the surface area of  S .

b) (16 marks) Let  F ( x , y , z ) = Ñ g ( x , y , z ) + H ( x , y , z ) , where  g ( x , y , z ) = e ^x

and  H ( x , y , z ) = e ^{y z} i + 6 x k . Compute the line integral of the vector field  F  along the path

c ( t ) = ( t ² , 1+ t , 1 – t ) ,  0 ≤ t ≤ 1 .      

3. (18 marks) Evaluate the double integral   , where  W   is the region

consisting of all the points  ( x , y )  such that  x ³ 0 ,  y ³ 0 ,  x ≤ y ²  and  4 ≤ x + y ² ≤ 6 .

Hint: Use the change of variables  x = u – v  and   .

4. (18 marks) Consider the unbounded solid region

W = { ( x , y , z ) ½  x ² + y ² + z ² ³ 1  and   }

with density given by the function  d  ( x , y , z ) = z ^{– 5} . Find the coordinates of the centre of mass of  W .

5. a) (7 marks) Consider the path  c ( t ) = ( 2 sin ² t ,  sin ³ t , cos ³ t ) ,  0 ≤ t ≤ π .

Suppose that  F  is a vector field that is continuous over the path  c  and such that

 ≤ M  when  0 ≤ t ≤ π .

Show that   ≤ 5 M  .

b) (5 marks) Suppose that  a > 0 ,  b > 0 ,  and  that  f : [ 0 , a ] ® [ 0 ,  b ]  denotes a  C ¹  function.

Let  S  be the surface generated by revolving the graph of the function  y = f ( x )  about the  x-axis, and let

g ( x , y , z ) =  . Show that   ≤  2 π a b .