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4. y
x
∇𝑓 = 3𝑖 + 4𝑗 =
5
2
∇𝑔
(
3
4
,
4
5
)
𝑥2
+𝑦2
= 1
∇𝑔 =
6
5
𝑖 +
8
5
𝑗

6. y
x
𝑥2
8
+
𝑦2
2
= 1
√2
2√2

33. If we integrate first
with respect to y, we
integrate along a
vertical line through
R and then integrate
from left to right to
include all the
vertical lines in R.

34. If we integrate first with
respect to x, we integrate
along a horizontal line
through R and then
integrate from bottom to top
to include all the horizontal
lines in R.