MAT368H5S
Lecture 1
Jan 8, 2013
Review of One Variable Integrals
If f : [a, b] R, f 0, then the denite integral of f on [a, b] can be dened
non-rigorously by
b
f (x)dx = Area below the graph of f .
a
Based on this we can come up with the rigorous denition
MAT368H5S
Lecture 14
Mar 5, 2013
Line Integrals in Two Dimensions (continued)
In the previous lecture we dened line integrals with respect to arc length, x, and y . Among them the
integral with respect to arc length is the most natural and has some nice p
MAT368H5S
Lecture 13
Feb 28, 2013
Vector Fields
A vector eld over a set E R2 /R3 is a function that assigns a 2/3-dimensional vector to each point of
E.
Any space of 2/3-dimensional vectors can be identied with R2 /R3 (the components of the vector are
ide
MAT368H5S
Lecture 12
Feb 14, 2013
Change of Variables (continued)
We already saw to use the change of variables formula when we integrated in polar, cylindrical, and
spherical coordinates. The applications of this formula go beyond these particular cases.
MAT368H5S
Lecture 11
Feb 12, 2013
Spherical Coordinates
Any point P (x, y, z ) R3 can be identied by (, , ), where 0 is the distance from P to the origin,
[0, 2 ] is the angle from the polar coordinates of the projection of P onto the xy -plane, and [0,
MAT368H5S
Lecture 10
Feb 7, 2013
We saw that cylindrical coordinates are useful for integrals over solids that can be obtained through
rotation around the z -axis. The cylindrical coordinates can be easily adjusted to work for solids obtained
through rota
MAT368H5S
Lecture 9
Feb 5, 2013
Example. Reverse the order of integration in the following integral
y2
1
1y
f (x, y, z )dzdxdy.
0
0
0
Solution. We rst sketch the solid E determined by the limits of integration. We can then switch the
order of integration
MAT368H5S
Lecture 8
Example. Evaluate
the plane x = 0.
E
Jan 31, 2013
y 2 z 2 dV , where E is the solid bounded by the paraboloid x = 1 y 2 z 2 and
Solution. We rst sketch E and its projection D onto the yz -plane (this means we chose to integrate x
rst).
MAT368H5S
Lecture 7
Jan 29, 2013
The following example illustrates what happens when we try to use polar coordinates on non-circular
regions.
Example. Evaluate
R
xy
x2 + y 2 dA, where R = [0, 1] [0, 1].
Solution. Looking at the integrand it is natural to
MAT368H5S
Lecture 6
Jan 24, 2013
Double Integrals in Polar Coordinates
To integrate in polar coordinates means to make the change of variables (x, y ) = T (r, ) = (r cos , r sin ).
To apply the change of variables formula we will need to know the determin
MAT368H5S
Lecture 5
Jan 22, 2013
Change of Variables in Double Integrals
Before we state and prove the change of variables formula, we need to discuss how area changes under a
linear transformation and recall the notion of linear approximation.
Change of
MAT368H5S
Lecture 4
Jan 17, 2013
Example. Find the volume of the solid bounded by the parabolic cylinder y = x2 and the planes z = 3y ,
z = 2 + y.
Solution.
Observe that the two planes intersect along the line z = 3, y = 1. Let D be the projection of the
MAT368H5S
Lecture 3
Jan 15, 2013
Double Integrals over General Sets
Let D R2 and f : D R. We can dene the double integral of f over D
in the same non-rigorous way as for the double integral over a rectangle:
f (x, y )dA = Signed volume between the graph o
MAT368H5S
Lecture 2
Jan 10, 2013
Example. Find the volume of the solid enclosed by the surface z = 1 + ex sin y and the planes x = 1,
y = 0, y = and z = 0.
Solution. The problem can be restated as follows: Find the volume of the solid between the graph of
MAT368H5S
Lecture 15
Mar 7, 2013
Example. Evaluate C 2y dx + 2z dy + 2x dz , where C is the intersection of the cylinder x2 + y 2 = 1
and the plane z = y + 1, in the rst octant, going from the xz plane to the yz plane.
Solution.
We rst parameterize C .
C: