Calc 3 Lecture Notes
Section 14.4
Page 1 of 6
Section 14.4: Green's Theorem
Big idea: Certain contour line integrals are equivalent to a double integral over the region
bounded by the contour. This is Green’s Theorem.
Big skill: You should be able to evaluate certain line integrals on a contour using Green’s
Theorem.
Vocabulary:
Curve C: C   x, y  | x  g  t  , y  h  t  , a  t  b
A curve C is called simple if it does not intersect itself except at the endpoints.
A simple closed curve C has positive orientation if the region R that it bounds “stays to the left”
of C as it is traversed.
The notation
 F  dr is used to denote a line integral along a simple closed curve C with a
C
positive orientation.
Calc 3 Lecture Notes
Section 14.4
Page 2 of 6
Theorem 4.1: Green’s Theorem
If C is a piecewise-smooth, simple closed curve in the plane with positive orientation, R is the
region enclosed by C together with C, M  x, y  and N  x, y  are continuous with continuous
first partial derivatives in some open region D, and R  D, then
 N M 
C M  x, y  dx  N  x, y  dy  R  x  y  dA .
Proof:
This proof restricts itself to the case that the region R is convex, meaning that we can represent R
as: C   x, y  | a  x  b, g1  x   y  g2  x    x, y  | c  y  d , h1  y   x  h2  y  . We then
integrate the dx and dy terms of the integral separately using these two representations of R.
Calc 3 Lecture Notes
Section 14.4
Page 3 of 6
Note that it is kind of surprising that an integral around the boundary of a region is equivalent to
a double integral over the entire region itself. However, we have seen examples of functions of
one variable where the behavior of the function over an interval was related to the behavior of
the function at its endpoints:
Theorem 4.4 (Intermediate Value Theorem):
If f is continuous on the closed interval [a, b] and W is any number between f(a) and f(b), then
there is at least one number c  [a, b] for which f(c) = W.
Theorem 9.4 (Mean Value Theorem):
If f(x) is continuous on the interval [a, b] and differentiable on the interval (a, b), then there is a
f (b)  f (a )
number c  (a, b) such that f ' (c) 
.
ba
Theorem 4.4: Integral Mean Value Theorem
b
1
f  x  dx .
b  a a
(i.e., a continuous function takes on its average value at some point in any given interval.)
If f is continuous on [a, b], then there is a number c  (a, b) for which f  c  
Calc 3 Lecture Notes
Section 14.4
Practice:
1. Use Green’s Theorem to evaluate
 x
2
Page 4 of 6
 y 3  dx  3xy 2 dy , where C encompasses the area
C
under the parabola y = x2 as shown in the picture below.
2. Use Green’s Theorem to evaluate
 7 y  e
C
a circle of radius 3 centered at (5, -7).
sin  x 
 dx  15x  sin  y
3

 8 y  dy , where C is
Calc 3 Lecture Notes
Section 14.4
3. Use Green’s Theorem to evaluate
 e
x
Page 5 of 6


 6 xy  dx  8 x 2  sin  y 2  dy , where C is shown
C
in the picture below.
Vocabulary:
R is the contour C that bounds a region R.
 N M 

Thus,  M  x, y  dx  N  x, y  dy   
 dA .
x y 
R
R 
Green’s Theorem also works over a bounded region that is not simply connected; you just
have to remember to do the line integral over the inner and outer boundary. The trick to showing
that Green’s Theorem still applies is to make a cut across the region. The line integrals
traversing the cut in opposite directions will cancel, leaving you with the line integrals you want.
Calc 3 Lecture Notes
Section 14.4
Page 6 of 6
Practice:
4. For F  x, y  
 y, x
, show that
x2  y 2
 F  dr  2 for every simple closed curve C enclosing
C
the origin.
Theorem 4.2:
Suppose M  x, y  and N  x, y  have continuous first partial derivatives on a simply-connected
region D. Then
 M  x, y  dx  N  x, y  dy is independent of path if and only if
C
M y  x, y   N x  x, y  for all (x, y) in D.