TEST3/MAC2313
Page 4 of 5
______________________________________________________________________
7.
(5 pts.)
Write down the triple iterated integral in spherical
coordinates that would be used to compute the volume of the solid
G
bounded
above by the sphere defined by
ρ
= 4 and below by the cone defined by
φ
=
π
/3.
Do not attempt to evaluate the iterated integrals.
Ouch!
It feels like an easy spherical wedgie.
______________________________________________________________________
8.
(15 pts.)
Let
f
(
x
,
y
) =
x
2
y
2
on the closed unit disk defined by
x
2
+
y
2
≤
1.
Find the absolute extrema and where they occur.
Use Lagrange multipliers to analyze the function on the
boundary.
Do not neglect the interior of the disk in doing your analysis!!
//
First, we deal with interior points of the disk.
Since
∇
f
(
x
,
y
) = < 2
xy
2
, 2
yx
2
>, there are infinitely many critical points in the
interior of the disk where either component of the pair (
x
,
y
) is zero.
Obviously,
f
(x,y) = 0 if either
x
= 0 or
y
= 0.
Now to study
f
on the boundary using Lagrange multipliers, set
g
(
x
,
y
) =
x
2
+
y
2
- 1.
Then (
x
,
y
) is on the circle defined by
x
2
+
y
2
= 1
precisely when (
x
,
y
) satisfies
g
(
x
,
y
) = 0.
Since
∇
g
(
x
,
y
) = <2
x
, 2
y
>,
∇
g
(
x
,
y
)
≠
<0, 0> when (
x
,
y
) is on the circle given by
g
(
x
,
y
) = 0.
Plainly
f
and
g
are smooth enough to satisfy the hypotheses of the Lagrange
Multiplier Theorem.
Thus, if a constrained local extremum occurs at (
x
,
y
),
there is a number
λ
so that
∇
f
(
x
,
y
) =
λ∇
g
(
x
,
y
).
Now
∇
f
(
x
,
y