1. Our goal in this problem is to ﬁnd the area of the region
D
bounded by the ellipse
x
2
a
2
+
y
2
b
2
= 1
,
for real numbers
a, b >
0.
a. (4 points) Find a linear change of coordinates
T
:
R
2
→
R
2
that maps the unit square
with coordinates (0
,
0), (1
,
0), (0
,
1), (1
,
1) to the rectangle with coordinates (0
,
0), (
a,
0),
(0
, b
), (
a, b
).
Solution:
T
(
u, v
) = (
au, bv
). In other words,
T
(
u, v
) =
(
x
(
u, v
)
, y
(
u, v
)
)
, where
x
(
u, v
) =
au
and
y
(
u, v
) =
bv.
b. (2 points) Describe the region
D
*
in
R
2
which maps to
D
under
T
.
Solution:
D
*
is the unit disk centered at the origin:
D
*
=
{
(
u, v
)

u
2
+
v
2
≤
1
}
.
c. (6 points) Express the area of
D
as a double integral over
D
*
, and compute this integral.
Solution: The Jacobian of the transformation