LESSON 15
Double Integrals Over General Regions
READ:
Section 16.2
NOTES:
Integrating a function of two variables over a rectangular
region in the
x
,
y
plane is not always sufficient for
applications. It might be necessary, for example, to compute an integral of the form
RR
D
f
(
x, y
)
dA
, where
D
is a circular disk in the
x
,
y
plane. The bad news is that, in general, there is no nice way to compute such
double integrals without going back to the definition of the integral as a limit of Riemann sums as described
in section 16.1, at least when the region has a very complicated boundary. The good news is that for most
types of regions you are likely to meet in real life, there is often an easy way to do the integral which is very
similar to the method used to evaluate a double integral over a rectangle. The idea is again to convert the
double integral into an iterated integral.
One type of region that can be handled is the area trapped between two functions of
x
, say
y
=
g
2
(
x
) and
y
=
g
1
(
x
) for
a
≤
x
≤
b
, as pictured in figure 5, page 869 of the text. Let’s call that region
D
. Suppose
z
=
f
(
x, y
) is the equation for a surface over the region
D
, and we want to compute
RR
D
f
(
x, y
)
dA
. The
reasoning of the last lesson can be repeated.
The plan is to compute the volume of the solid under the
surface above the region
D
by first finding a formula for the area of the crosssections produced when the
planes
x
=
c
intersects the solid, and then integrate the crosssections for all the
x
’s from
a
to
b
.
If the solid is sliced by the plane, as described above, through a particular
x
, the crosssection will be a
plane region bounded by the curve
z
=
f
(
x, y
) on the top (remember,
x
is fixed
there, and
y
is the variable
in
f
(
x, y
)), with the
y
values in the interval
g
1
(
x
) to
g
2
(
x
). That is the key for doing the double integral. In
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 Winter '11
 JerryMetzger
 Calculus, Integrals, Region, Iterated Integrals

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