9 the constrained extrema of fxy subject to the

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 .