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AP Calculus AB
Unit 8: Applications of Integration
8.2 Finding the Area Between Curves Expressed as Functions of <latex>y</latex>
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What is the formula to find the area between curves expressed as functions of
y
y
y
?
A
r
e
a
=
Area =
A
re
a
=
∫
a
b
[
f
(
y
)
−
g
(
y
)
]
d
y
\int_{a}^{b} [f(y) - g(y)] \, dy
∫
a
b
[
f
(
y
)
−
g
(
y
)]
d
y
When finding the area between curves, it is necessary to ensure that
f
(
y
)
≥
g
(
y
)
f(y) \geq g(y)
f
(
y
)
≥
g
(
y
)
over the interval
[
a
,
b
]
[a, b]
[
a
,
b
]
.
True
For
y
∈
[
0
,
1
]
y \in [0, 1]
y
∈
[
0
,
1
]
,
2
y
−
y
2
≥
y
2
2y - y^{2} \geq y^{2}
2
y
−
y
2
≥
y
2
.
True
The intersection points between the curves
x
=
x =
x
=
y
2
y^{2}
y
2
and x = 2y - y^{2}</latex> occur at
y
=
y =
y
=
0
0
0
and
y
=
y =
y
=
1
1
1
.
True
Which curve is on the right in the example
x
=
x =
x
=
2
y
−
y
2
2y - y^{2}
2
y
−
y
2
and
x
=
x =
x
=
y
2
y^{2}
y
2
for
y
∈
[
0
,
1
]
y \in [0, 1]
y
∈
[
0
,
1
]
?
2
y
−
y
2
2y - y^{2}
2
y
−
y
2
The y</latex> values of the intersection points are used as the limits of integration for finding the area between curves expressed as functions of
y
y
y
.
True
What is the method to identify the upper and lower curves when expressed as functions of
y
y
y
?
Compare
x
x
x
values
What is the integrand when
2
y
−
y
2
2y - y^{2}
2
y
−
y
2
is the upper curve and y^{2}</latex> is the lower curve?
2
y
−
2
y
2
2y - 2y^{2}
2
y
−
2
y
2
The formula to find the area between curves expressed as functions of
y
y
y
is
A
r
e
a
=
Area =
A
re
a
=
∫
a
b
[
f
(
y
)
−
g
(
y
)
]
d
y
\int_{a}^{b} [f(y) - g(y)] \, dy
∫
a
b
[
f
(
y
)
−
g
(
y
)]
d
y
.
True
To determine the limits of integration, find the
y
y
y
values where the curves intersect.
True
To determine the limits of integration for area between curves expressed as functions of
y
y
y
, the first step is to express the curves as functions
Expressing curves as functions of
y
y
y
involves writing them in the form x = f(y)</latex> and
x
=
x =
x
=
g
(
y
)
g(y)
g
(
y
)
to find the area between their intersection
Steps to determine the limits of integration for area between curves expressed as functions of
y
y
y
1️⃣ Express the curves as
x
=
x =
x
=
f
(
y
)
f(y)
f
(
y
)
and
x
=
x =
x
=
g
(
y
)
g(y)
g
(
y
)
2️⃣ Equate
f
(
y
)
=
f(y) =
f
(
y
)
=
g
(
y
)
g(y)
g
(
y
)
to find
y
y
y
values
3️⃣ Use the
y
y
y
values as the limits
a
a
a
and
b
b
b
The integrand for finding the area between two curves expressed as functions of
y
y
y
is the difference between their
y
y
y
values.
False
The antiderivative of
2
y
−
2
y
2
2y - 2y^{2}
2
y
−
2
y
2
is
y
2
−
2
3
y
3
+
y^{2} - \frac{2}{3}y^{3} +
y
2
−
3
2
y
3
+
C
C
C
.
True
In the formula for finding the area between curves,
f
(
y
)
f(y)
f
(
y
)
represents the right curve, while
g
(
y
)
g(y)
g
(
y
)
represents the left curve
What are the curves used in the example to find the area between them?
x
=
x =
x
=
y
2
y^{2}
y
2
and
x
=
x =
x
=
2
y
−
y
2
2y - y^{2}
2
y
−
y
2
What is the integral to find the area between the curves
x
=
x =
x
=
y
2
y^{2}
y
2
and x = 2y - y^{2}</latex>?
A
r
e
a
=
Area =
A
re
a
=
∫
0
1
(
2
y
−
2
y
2
)
d
y
\int_{0}^{1} (2y - 2y^{2}) \, dy
∫
0
1
(
2
y
−
2
y
2
)
d
y
In the formula to find the area between curves expressed as functions of
y
y
y
, what does
f
(
y
)
f(y)
f
(
y
)
represent?
Right curve
Steps to determine the limits of integration for area between curves expressed as functions of
y
y
y
.
1️⃣ Set up the curves as functions of
y
y
y
2️⃣ Find the intersection points
3️⃣ Use the
y
y
y
values as the limits of integration
Steps to identify the upper and lower curves expressed as functions of
y
y
y
.
1️⃣ Compare the curves over the interval
2️⃣ The curve with the larger
x
x
x
value is the upper curve
3️⃣ The curve with the smaller
x
x
x
value is the lower curve
The curve with the larger
x
x
x
value for a given
y
y
y
is the upper curve.
True
The integrand for finding the area between the curves
x
=
x =
x
=
2
y
−
y
2
2y - y^{2}
2
y
−
y
2
and
x
=
x =
x
=
y
2
y^{2}
y
2
is
2
y
−
2
y
2
2y - 2y^{2}
2
y
−
2
y
2
.True
In the area formula, what does
f
(
y
)
f(y)
f
(
y
)
represent?
Right curve
What step follows setting up the curves as functions of
y
y
y
to find the limits of integration?
Find intersection points
Finding the intersection points of two curves expressed as functions of
y
y
y
involves equating their
f
(
y
)
f(y)
f
(
y
)
and
g
(
y
)
g(y)
g
(
y
)
values.
True
The formula for finding intersection points of curves expressed as functions of
y
y
y
is
f
(
y
)
=
f(y) =
f
(
y
)
=
g
(
y
)
g(y)
g
(
y
)
.
True
Match the curve type with its description:
Upper curve ↔️ Curve with larger
x
x
x
values for a given
y
y
y
Lower curve ↔️ Curve with smaller
x
x
x
values for a given
y
y
y
What is the integrand for the curves
x
=
x =
x
=
2
y
−
y
2
2y - y^{2}
2
y
−
y
2
(upper) and
x
=
x =
x
=
y
2
y^{2}
y
2
(lower)?
2
y
−
2
y
2
2y - 2y^{2}
2
y
−
2
y
2
What is the area between the curves
x
=
x =
x
=
y
2
y^{2}
y
2
and x = 2y - y^{2}</latex> over the interval
[
0
,
1
]
[0, 1]
[
0
,
1
]
?
1
3
\frac{1}{3}
3
1
Steps to find the area between curves expressed as functions of
y
y
y
1️⃣ Determine the curves and bounds
2️⃣ Identify the right and left curves
3️⃣ Set up the integral
4️⃣ Evaluate the integral to find the area
In the example, the limits of integration are
a
=
a =
a
=
0
0
0
and b = 1
Match the step with its description in determining the limits of integration:
Curves as functions of
y
y
y
↔️
f
(
y
)
=
f(y) =
f
(
y
)
=
2
y
−
y
2
2y - y^{2}
2
y
−
y
2
,
g
(
y
)
=
g(y) =
g
(
y
)
=
y
2
y^{2}
y
2
Intersection points ↔️
2
y
−
y
2
=
2y - y^{2} =
2
y
−
y
2
=
y
2
y^{2}
y
2
Limits of integration ↔️
a
=
a =
a
=
0
0
0
,
b
=
b =
b
=
1
1
1
When finding the area between curves expressed as functions of
y
y
y
, it is necessary to ensure
f
(
y
)
≥
g
(
y
)
f(y) \geq g(y)
f
(
y
)
≥
g
(
y
)
over the interval
[
a
,
b
]
[a, b]
[
a
,
b
]
.
True
What are the
y
y
y
values where
f
(
y
)
=
f(y) =
f
(
y
)
=
2
y
−
y
2
2y - y^{2}
2
y
−
y
2
and
g
(
y
)
=
g(y) =
g
(
y
)
=
y
2
y^{2}
y
2
intersect?
0 and 1
Which curve has the larger
x
x
x
value for a given
y
y
y
in the interval [0, 1]</latex>?
2
y
−
y
2
2y - y^{2}
2
y
−
y
2
For
0
≤
y
≤
1
0 \leq y \leq 1
0
≤
y
≤
1
, the upper curve is
f
(
y
)
=
f(y) =
f
(
y
)
=
2
y
−
y
2
2y - y^{2}
2
y
−
y
2
, since
2
y
−
y
2
≥
y
2
2y - y^{2} \geq y^{2}
2
y
−
y
2
≥
y
2
.True
How do you write the integrand as the difference between the upper and lower curves?
Subtract lower from upper
For the curves
x
=
x =
x
=
2
y
−
y
2
2y - y^{2}
2
y
−
y
2
and
x
=
x =
x
=
y
2
y^{2}
y
2
, the limits of integration are
a
=
a =
a
=
0
0
0
and
b
=
b =
b
=
1
1
1
.True
What values are used as the limits of integration for curves expressed as functions of
y
y
y
?
y
y
y
values
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