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AP Calculus BC
Unit 8: Applications of Integration
8.10 Volume with Disc Method: Revolving Around Other Horizontal or Vertical Lines
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What is the radius formula when revolving around a horizontal line
y
=
y =
y
=
k
k
k
?
∣
f
(
x
)
−
k
∣
|f(x) - k|
∣
f
(
x
)
−
k
∣
What is the radius formula when revolving around a vertical line
x
=
x =
x
=
h
h
h
?
∣
g
(
y
)
−
h
∣
|g(y) - h|
∣
g
(
y
)
−
h
∣
Match the axis of revolution with its corresponding radius and volume formulas:
y
=
y =
y
=
k
k
k
(Horizontal) ↔️
∣
f
(
x
)
−
k
∣
|f(x) - k|
∣
f
(
x
)
−
k
∣
,
∫
a
b
π
(
f
(
x
)
−
k
)
2
d
x
\int_{a}^{b} \pi (f(x) - k)^{2} dx
∫
a
b
π
(
f
(
x
)
−
k
)
2
d
x
x
=
x =
x
=
h
h
h
(Vertical) ↔️
∣
g
(
y
)
−
h
∣
|g(y) - h|
∣
g
(
y
)
−
h
∣
,
∫
c
d
π
(
g
(
y
)
−
h
)
2
d
y
\int_{c}^{d} \pi (g(y) - h)^{2} dy
∫
c
d
π
(
g
(
y
)
−
h
)
2
d
y
When revolving
f
(
x
)
=
f(x) =
f
(
x
)
=
x
2
x^{2}
x
2
around
y
=
y =
y
=
−
1
- 1
−
1
, the volume is \frac{286\pi}{15}
The disc method is used to calculate the volume of a solid of
revolution
When revolving around a horizontal line y = k</latex>, the volume formula is
∫
a
b
π
(
f
(
x
)
−
k
)
2
d
x
\int_{a}^{b} \pi (f(x) - k)^{2} dx
∫
a
b
π
(
f
(
x
)
−
k
)
2
d
x
.
True
When revolving around a vertical line x = h</latex>, the volume formula is
\int_{c}^{d}
What is the radius when revolving
f
(
x
)
=
f(x) =
f
(
x
)
=
x
2
x^{2}
x
2
around the line
y
=
y =
y
=
−
1
- 1
−
1
?
x
2
+
x^{2} +
x
2
+
1
1
1
The axis of rotation determines the shape and radius of the discs in the
disc method
.
True
What is the radius when revolving
f
(
x
)
=
f(x) =
f
(
x
)
=
x
2
x^{2}
x
2
around the line
y
=
y =
y
=
2
2
2
?
∣
x
2
−
2
∣
|x^{2} - 2|
∣
x
2
−
2∣
What direction is the radius when revolving around a horizontal axis?
Perpendicular to x-axis
When revolving around a horizontal axis
y
=
y =
y
=
k
k
k
, the radius is |f(x) - k|
What is the first step when calculating volumes using the disc or washer method?
Identify the axis of rotation
When revolving around a vertical line
x
=
x =
x
=
h
h
h
, the radii of the discs are perpendicular to the y-axis.
If
f
(
x
)
=
f(x) =
f
(
x
)
=
x
2
x^{2}
x
2
and the axis of rotation is
y
=
y =
y
=
2
2
2
, what is the radius of the disc?
∣
x
2
−
2
∣
|x^{2} - 2|
∣
x
2
−
2∣
How is the radius of each disc determined when revolving around a horizontal axis
y
=
y =
y
=
k
k
k
?
∣
f
(
x
)
−
k
∣
|f(x) - k|
∣
f
(
x
)
−
k
∣
Match the axis of rotation with its radius formula:
Horizontal (
y
=
y =
y
=
k
k
k
) ↔️
∣
f
(
x
)
−
k
∣
|f(x) - k|
∣
f
(
x
)
−
k
∣
Vertical (
x
=
x =
x
=
h
h
h
) ↔️
∣
g
(
y
)
−
h
∣
|g(y) - h|
∣
g
(
y
)
−
h
∣
Steps to set up the integral for calculating volume using the disc method:
1️⃣ Identify the axis of rotation
2️⃣ Determine the radius formula
3️⃣ Write the volume formula
What is the volume formula for a horizontal axis of rotation
y
=
y =
y
=
k
k
k
?
∫
a
b
π
(
f
(
x
)
−
k
)
2
d
x
\int_{a}^{b} \pi (f(x) - k)^{2} dx
∫
a
b
π
(
f
(
x
)
−
k
)
2
d
x
Match the axis of rotation with its volume formula:
y
=
y =
y
=
k
k
k
(Horizontal) ↔️
∫
a
b
π
(
f
(
x
)
−
k
)
2
d
x
\int_{a}^{b} \pi (f(x) - k)^{2} dx
∫
a
b
π
(
f
(
x
)
−
k
)
2
d
x
x
=
x =
x
=
h
h
h
(Vertical) ↔️
∫
c
d
π
(
g
(
y
)
−
h
)
2
d
y
\int_{c}^{d} \pi (g(y) - h)^{2} dy
∫
c
d
π
(
g
(
y
)
−
h
)
2
d
y
The volume formula for revolving around a vertical axis
x
=
x =
x
=
h
h
h
is \int_{c}^{d} \pi (g(y) - h)^{2} dy</latex>.
What three components are essential for calculating volumes using the disc method?
Axis, radius, volume formula
What is the formula for the radius when revolving around a horizontal axis
y
=
y =
y
=
k
k
k
?
|f(x) - k|</latex>
When revolving around a horizontal axis
y
=
y =
y
=
k
k
k
, the volume formula is \int_{a}^{b} \pi (f(x) - k)^{2} dx</latex>.volume
Match the axis of rotation with the direction of the radius and the form of the function:
y
=
y =
y
=
k
k
k
(Horizontal) ↔️ Perpendicular to x-axis,
f
(
x
)
f(x)
f
(
x
)
x
=
x =
x
=
h
h
h
(Vertical) ↔️ Perpendicular to y-axis,
g
(
y
)
g(y)
g
(
y
)
When revolving
g
(
y
)
=
g(y) =
g
(
y
)
=
y
\sqrt{y}
y
around
x
=
x =
x
=
−
1
- 1
−
1
, the radius is \sqrt{y} + 1
When revolving around a vertical axis, the radius is perpendicular to the y-axis.
True
What is the radius when revolving
g
(
y
)
=
g(y) =
g
(
y
)
=
y
\sqrt{y}
y
around the line
x
=
x =
x
=
−
1
- 1
−
1
?
∣
y
+
|\sqrt{y} +
∣
y
+
1
∣
1|
1∣
When revolving around a horizontal line y = k</latex>, the radii of the discs are perpendicular to the x-axis.
True
Match the axis of rotation with the direction of the radius:
y
=
y =
y
=
k
k
k
↔️ Perpendicular to x-axis
x
=
x =
x
=
h
h
h
↔️ Perpendicular to y-axis
If
g
(
y
)
=
g(y) =
g
(
y
)
=
y
\sqrt{y}
y
and the axis of rotation is
x
=
x =
x
=
−
1
- 1
−
1
, the radius of the disc is
y
+
\sqrt{y} +
y
+
1
1
1
.
For a vertical axis
x
=
x =
x
=
h
h
h
, the radius formula is
∣
g
(
y
)
−
h
∣
|g(y) - h|
∣
g
(
y
)
−
h
∣
.
True
If f(x) = x^{2}</latex> is revolved around
y
=
y =
y
=
2
2
2
, what is the radius of the disc?
∣
x
2
−
2
∣
|x^{2} - 2|
∣
x
2
−
2∣
For a horizontal axis of rotation
y
=
y =
y
=
k
k
k
, the radius formula is
∣
f
(
x
)
−
k
∣
|f(x) - k|
∣
f
(
x
)
−
k
∣
.
What is the volume formula for a vertical axis of rotation
x
=
x =
x
=
h
h
h
?
∫
c
d
π
(
g
(
y
)
−
h
)
2
d
y
\int_{c}^{d} \pi (g(y) - h)^{2} dy
∫
c
d
π
(
g
(
y
)
−
h
)
2
d
y
The volume formula for revolving around a horizontal axis
y
=
y =
y
=
k
k
k
is
∫
a
b
π
(
f
(
x
)
−
k
)
2
d
x
\int_{a}^{b} \pi (f(x) - k)^{2} dx
∫
a
b
π
(
f
(
x
)
−
k
)
2
d
x
.
True
When using the disc method, the axis of rotation can be either a horizontal line
y
=
y =
y
=
k
k
k
or a vertical line
x
=
x =
x
=
h
h
h
.rotation
When revolving around a vertical axis
x
=
x =
x
=
h
h
h
, the radius formula is
∣
g
(
y
)
−
h
∣
|g(y) - h|
∣
g
(
y
)
−
h
∣
.
True
When revolving around a vertical axis
x
=
x =
x
=
h
h
h
, the volume formula is
∫
c
d
π
(
g
(
y
)
−
h
)
2
d
y
\int_{c}^{d} \pi (g(y) - h)^{2} dy
∫
c
d
π
(
g
(
y
)
−
h
)
2
d
y
.
True
Steps to calculate volume using the disc method
1️⃣ Identify the axis of rotation
2️⃣ Determine the radius formula
3️⃣ Write the volume formula
4️⃣ Calculate the integral
See all 62 cards