PRINT NAME: SOLUTIONS
Calculus IV [2443–002] Midterm III
1
Q1 [15 points]
1.1
Part 1
Write down the change of variables formula for triple integrals.
1.2
Answer to part 1
Suppose the change of variables (
x
(
u,v,w
)
,y
(
u,v,w
)
,z
(
u,v,w
)) takes a region
S
in
uvw
-space to a region
R
in
xyz
-space. Then
ZZZ
R
f
(
x,y,z
)
dV
=
ZZZ
S
f
(
x
(
u,v,w
)
,y
(
u,v,w
)
,z
(
u,v,w
))
±
±
±
±
±
∂
(
x,y,z
)
∂
(
u,v,w
)
±
±
±
±
±
dudv dw
where
∂
(
x,y,z
)
∂
(
u,v,w
)
=
±
±
±
±
±
±
±
x
u
x
v
x
w
y
u
y
v
y
w
z
u
z
v
z
w
±
±
±
±
±
±
±
1.3
Part 2
Use the change of variables formula to evaluate the volume of the ellipsoid bounded by
x
2
a
2
+
y
2
b
2
+
z
2
c
2
= 1
Show all the steps of your work clearly. You may use the fact that the volume of a sphere of radius
r
is
equal to
4
3
πr
3
.
1.4
Answer to part 2
Recall that
Volume =
ZZZ
ellipsoid
dV
Let
x
=
au
,
y
=
bv
, and
z
=
cw
. Then the ellipsoid above becomes a unit ball bounded by the sphere
u
2
+
v
2
+
w
2
= 1 in
uvw
-space. We also have
x
u
=
a
,
x
v
=
x
w
= 0,
y
v
=
a
,
y
u
=
y
w
= 0, and
z
w
=
a
,
z
u
=
z
v
= 0. Thus we get
∂
(
x,y,z
)
∂
(
u,v,w
)
=
±
±
±
±
±
±
±
a
0 0
0
b
0
0 0
c
±
±
±
±
±
±
±
=
abc
and, substituting into the change of variables formula yields
ZZZ
ellipsoid
dV
=
ZZZ
unit ball
abcdudv dw
=
abc
(Volume of unit ball) =
4
π
3
abc.