9.
The constrained extrema of f(x,y) subject to the constraint g(x,y) = 0 will occur at a
point
where
f (x
0
, y
0
)
=
λ .
g(x
0
, y
0
)
provided
g(x
0
, y
0
) is not
0
.
We must also check the
endpoints of g(x,y) = 0; and at the points where
g=
0
or
f =
0
, if these lie on g(x,y) = 0.
11.
∫∫
R
f(x,y) dA
=
∫
[a,b]
{
∫
[g(x), h(x)]
f(x,y).dy
}
dx
=
∫
[c,d]
{
∫
[p(y), q(y)]
f(x,y).dx
}
dy
∫∫
R
f(x,y) dA
=
∫
[α,β]
{
∫
[g(θ), h(θ)]
f(r,θ) . r.dr
}
dθ
12.
Volume of the solid determined by z
1
(x,y) & z
2
(x,y) over the plane region R is given by
Volume(G) =
∫∫
R
{
z
2
(x,y) - z
1
(x,y)
}
.dA .
Area (R) =
∫∫
R
1
.dA
.
13.
Surface Area of the surface
z = f(x,y) is given by:
S
=
∫∫
R
√
{(f
x
)
2
+ (f
y
)
2
+ 1}. dA
Surface Area of the surface
r
=
x(u,v), y(u,v), z(u,v)
is:
∫∫
R
||(
r
/
u)×(
r
/
v)|| . dA
14.
Mass
of the solid region G with density f(x,y,z) at
x, y, z
is given by.
Mass(G) =
∫∫∫
G
f(x,y,z)
.dV .
Volume (G) =
∫∫∫
G
1
.dV .
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- Spring '06
- GRANTCHAROV
- Calculus, Multivariable Calculus, x,y
-
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