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Simon Fraser University
MATH 251
Summer 2004
Final Examination
Instructor: A. Belshaw Date: August 4, 2004
Name:
Student number:
Signature:
Instructions
1. DO NOT OPEN THIS BOOKLET UNTIL YOU ARE TOLD TO DO
SO.
- Fill in the information above.
- Have your student ID card showing on the desk.
- This booklet contains 11 printed pages in addition to this cover page.
- Do all your work in this test booklet. Show all your work. Use the back of the previous page if necessary.
- No book, paper, or device should be within reach.
- Students observed writing anything after the call to stop writing will be subject to summary penalties.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Total
[6] 1. Calculate the length of the curve parametrized by
r(t) = 〈t,
t
(^32) ,
t^2 2
from t = 0 to t = 2.
[4] 2. Find the equation of the surface generated by revolving the curve y = e−z^ around the z-axis.
[8] 4. Using the definition of curvature and a general parametric equation of a line in three dimensions, prove that the curvature of a straight line in space is zero.
- Given f (x, y, z) =
xyz x^2 + y^2 + z^2
[6] ,
prove, using spherical coordinates, whether or not
lim (x,y,z)→(0, 0 ,0)
f (x, y, z)
exists. If the limit exists, find it.
[5] 6. Use a double integral to find the volume bounded above by the paraboloid
x^2 + y^2 +
z 2
= 1 and below by the xy-plane.
[8] 8. Find the maximum value of f (x, y, z) = xyz, on the intersection of the sphere x^2 + y^2 + z^2 = 1 and the plane z = 1/2, using the method of Lagrange multipliers.
[6] 9. Use triple integration to find the moment of inertia around the z-axis of the solid bounded above by z = 3 and below by the paraboloid z = 2x^2 + 2y^2. Assume density δ ≡ 1.
[6] 10. Find the surface area of the surface parametrized by r(u, v) = 〈sin u, cos u, v〉 for 0 ≤ u ≤ 2 π and 1 ≤ v ≤ 2.
[5] 12. Prove that curl(gradf ) = 0 , if the real-valued function f (x, y, z) has continuous second-order partial derivatives.
[6] 13. Use the vector form of Green’s theorem to calculate the flux of the field F(x, y) = 〈x^2 , y^2 〉 across the circle x^2 + y^2 = 4.
[8] 14. Find the centroid of a half-turn of wire of uniform density given by r(t) = 〈5 cos t, 5 sin t, t〉, letting t run from 0 to π. Show either that the centroid is on the wire or that it is not on the wire.
