Multivariable Calculus Quiz
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MULTIPLE INTEGRALS
Question 1
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The iterated integral
∫
0
1
∫
y
y
f
(
x
,
y
)
d
x
d
y
\int_0^1 \int_y^{\sqrt{y}} f(x,y) \, dx \, dy
∫
0
1
∫
y
y
f
(
x
,
y
)
d
x
d
y
represents integration over a region
R
R
R
. Which of the following correctly describes this region?
A
The region bounded by
y
=
x
2
y = x^2
y
=
x
2
,
x
=
y
2
x = y^2
x
=
y
2
, and the lines
x
=
0
x = 0
x
=
0
and
y
=
1
y = 1
y
=
1
B
The region where
0
≤
y
≤
1
0 \leq y \leq 1
0
≤
y
≤
1
and
y
≤
x
≤
y
y \leq x \leq \sqrt{y}
y
≤
x
≤
y
, which exists only when
y
≥
1
y \geq 1
y
≥
1
C
The region bounded by
y
=
x
2
y = x^2
y
=
x
2
,
x
=
y
x = \sqrt{y}
x
=
y
, with
0
≤
y
≤
1
0 \leq y \leq 1
0
≤
y
≤
1
D
The region is empty because
y
≤
y
y \leq \sqrt{y}
y
≤
y
only when
y
≥
1
y \geq 1
y
≥
1
, but we have
0
≤
y
≤
1
0 \leq y \leq 1
0
≤
y
≤
1
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