UNIVERSITY OF TORONTO
DEPARTMENT OF MATHEMATICS
MAT 237 Y - MULTIVARIABLE CALCULUS
ASSIGNMENT #6. DUE ON MARCH 16, 2006.
1. a) Suppose that a thin wire is bent into the shape on the (unbounded) curve of intersection of the surfaces
x = 5 y 2 and z = 7 y . Find the mass of the wire if its linear density at any point ( x , y , z ) is given by the
function d ( x , y , z ) = ( 1 + 2 y 2 ) – 3 / 2 .
b) Evaluate the line integral , where C is the curve
and z = t 2 , with 0 £ t £ 2 . Use your result to compute .
2. a) Let F ( x , y , z ) = ( 2 x z + y e x + y ) i + ( e x + z e y z ) j + y e y z k and let C be the curve
x = sin t , y = 2 cos t and z = sin ( 2 t ) , with 0 ≤ t ≤ 2 p . Compute .
b) Let F ( x , y , z ) = i + j + k and let C
be the curve x = t , y = e t , z = e 2 t , with 0 ≤ t ≤ 1 . Show that < 3 e 2 .
3. a) Evaluate the surface integral where S is the plane x + 2 y + 2 z = 6 .
b) Suppose that S is a right-circular conical shell with uniform density. Let h denote its height and let a denote the radius of its base. What happens to the centre of mass of the surface S when the value of a increases if the value of h remains unchanged?
4. Show that the value of the line integral does not depend on the parametrization of the curve C .