SURFACE AREA.
Today we learn how to evaluate the surface area. We assume that the
surface is given as a graph of function z = f (x, y) and the domain of this
function is a region D. We have
Theorem. The area of the surface given as a graph of the function

Assignment 1
1. If R = [-1, 3] X [0, 2], use a Riemann sum with m = 4, n = 2 to estimate the value of
(y2 2x2)dA . Take the sample points to be the upper left corners of the rectangles.
R
2.
a) Estimate the volume of the solid that lies below the surface

Math 263 Assignment 9 - Solutions
1. Find the ux of F = (x2 + y 2 )k through the disk of radius 3 centred at the origin in the xy
plane and oriented upward.
Solution The unit normal vector to the surface is n = k. The ux is thus given by:
F .dS =
x2 + y 2

Math 209
Assignment 8 Solutions
1. Use Greens Theorem to evaluate the line integral along the given positively oriented curve.
(a) C (y + e x )dx + (2x + cos y 2 )dy, C is the boundary of the region enclosed by the
parabolas y = x2 and x = y 2 .
Solution:

Homework - Section 16.8
MATH2263 - Summer 2010
1. 16.8 #9: Use Stokes Theorem to evaluate
C
F dr, where
F(x, y, z) = yz i + 2xz j + exy k
and C is the circle x2 + y 2 = 16, z = 5, oriented counterclockwise as seen from above.
Ans: The curve C is the bound

THE UNIVERSITY OF WESTERN ONTARIO
Department of Applied Mathematics
Name:
Student Number:
CALCULUS 2503b Midterm Examination
Thursday, Feb,16, 2012
Time: 5:45pm - 7:45pm
There are 7 questions; each question is worth 14 points.
Plus you get two points (o

Winter 2012
Math 255
Problem Set 11
Solutions
1) Dierentiate the two quantities with respect to time, use the chain
rule and then the rigid body equations.
17.6.18 Find a parametric representation for the surface which is the lower
half of the ellipsoid 2

THE UNIVERSITY OF WESTERN ONTARIO
Department of Applied Mathematics
Last Name:
First Name:
Student Number:
CALCULUS 2503b Make-Up Midterm Examination
Thursday, Feb,18, 2011
Time: 1:00pm - 4:00pm
There are 10 questions; each question is worth 10 points.

Quiz 21: Solutions
Math 2263:
Multivariable Calculus
Show all work!
Name:
Date:
1. 16.9 #5: Use the Divergence Theorem to calculate the surface integral
where
F(x, y, z) = ex sin y i + ex cos y j + yz 2 k,
S
F dS,
and where S is the surface of the box bou

Solutions - Homework sections 17.7-17.9
17.7
6.
Evaluate
S
xy dS, where S is the triangle with vertices (1, 0, 0), (0, 2, 0), and (0, 0, 2).
The three points - and therefore the triangle between them - are on the plane 2x + y + z = 2, and
their projection

THE UNIVERSITY OF WESTERN ONTARIO
Department of Applied Mathematics
First Name:
Last Name:
Student Number:
CALCULUS 2503b Final Examination
Thursday, April 28, 2011
Time: 7:00pm - 10:00pm
There are 10 questions with each question worth 10 points. Text boo

THE UNIVERSITY OF WESTERN ONTARIO
Department of Applied Mathematics
Last Name:
First Name:
Student Number:
CALCULUS 2503b Final Examination
Tuesday, April 24, 2012
Time: 7:00pm - 10:00pm
There are 10 questions and each is worth 10 points, although not all