Direct Calculation Methods for Integrals
1.
Denite Integral:
b
af(x)dx
. Here
a
and
b
could involve other variables besides
x
. Use Calculus I and II
to solve these.
2.
Line Integral:
CM dx +N dy +P dz
or
Cf ds
. Use this three-step procedure:
(a) Parametrize
C
in terms of
t
, say, with
a≤t≤b
.
(b) Compute dierentials
dx
,
dy
,
dz
,
ds
as needed.
(c) Reduce line integral to a denite integral on interval
a≤t≤b
by substitution.
3.
Iterated Integral:
b
a
g(v)
f(v)F(u, v)du dv
or
b
a
g(w)
f(w)
h(v,w)
g(v,w)F(u, v, w )du dv dw
. These are just denite
integrals two or three times.
4.
Double or Triple Integral:
Rf dA
or
Df dV
. Use a Fubini-type theorem to reduce these to iterated
integrals.
5.
Surface Integrals:
Sf dσ
. Use this three-step procedure:
(a) Parametrize
S
in terms of
u, v
, say, with
(u, v)∈R
, a region in the
uv
-plane and write out a position
vector
r(u, v) = hx(u, v), y (u, v), z (u, v )i
for points on
S
.
(b) Compute dierential
dσ =|ru×rv|dA
, where
dA
is dierential area in
uv
-plane.
Important special case where this is precomputed: If
S
is the graph of
z=f(x, y)
, then
dσ =
qf2
x+f2
y+ 1 dA
, where
dA
is dierential area in the
xy
-plane.
(c) Reduce surface integral to a double integral over
R
by substitution.
6.
Flux Integral:
SF·ndσ
. Here
n
is a continuous unit normal vector to
S
. Use same three-step procedure
as in 5, except that in (b)
ndσ =±ru×rvdA
.
Important special case where this is precomputed: If
S
is the graph of
z=f(x, y)
, then
ndσ =
± h−fx,−fy,1idA
, where
dA
is dierential area in the
xy
-plane.
Indirect Calculation Methods for Integrals
The idea is to indirectly calculate one side by directly calculating the other side of one of the following theorems.
1. (Flux integrals)
Gauss Divergence Theorem
in 3-D:
S
F·ndσ =
D
∇ · FdV
Here the boundary of the solid
D
is the (closed) surface
S=∂D
with outward pointing normal
n
.
2. Specialize the Divergence Theorem to 2-D in
xy
-plane and obtain the
ux form of Green's Theorem
for boundary closed curve
C=∂R
positively oriented with respect to the region
R
in the
xy
-plane:
C
F·nds =
C
M dy −N dx =
R
(Mx+Ny)dA =
R
∇ · FdA
3. (Flow Integrals)
Stokes' Theorem
:
C
F·dr=
S
∇ × F·ndσ
Here
C=∂S
is the closed boundary of the surface
S
, positively oriented with respect to the surface
normal
n
.
4. Specialize Stokes' Theorem to 2-D in
xy
-plane and obtain the
ow form of Green's Theorem
for
boundary closed curve
C=∂R
positively oriented with respect to the region
R
in the
xy
-plane:
C
F·dr=
C
M dx +N dy =
R
(Nx−My)dA =
R
∇ × F·kdA