The volume of each disk in the disk method is given by \( V = \pi r^2 h \), where \( r \) is the radius and \( h \) is the thickness
For rotation around the x-axis, the radius is \( r = f(x) \) and the height is \( h = \)dx
Match the axis of rotation with the corresponding variables and height:
x-axis ↔️ Variables: x, Height: dx
y-axis ↔️ Variables: y, Height: dy
What is the axis of revolution when a region is rotated to form a solid of revolution?
Line of rotation
The volume when rotating the region bounded by \( x = y^2 \), \( y = 0 \), and \( y = 2 \) around the y-axis is \( V = \int_{0}^{2} \pi (y^2)^2 dy = \pi \int_{0}^{2} \)y^4
For rotation around the x-axis, the volume is calculated using the integral \( V = \int_{a}^{b} \pi [f(x)]^2 dx \), where \( f(x) \) represents the radius
The volume of the solid formed by rotating the region bounded by \( y = \sqrt{x} \), \( x = 0 \), and \( x = 4 \) around the x-axis is \( V = \int_{0}^{4} \pi (\sqrt{x})^2 dx \).
True
When rotating around the y-axis, the radius for \( x = y^2 \) is \( r = y^2 \).
True
For the region bounded by \( y = x^2 \), \( y = 0 \), and \( x = 2 \), the limits of integration for rotation around the x-axis are from 0 to 2.
True
Steps to set up integrals using the disk method
1️⃣ Identify the axis of revolution
2️⃣ Determine the radius of each disk
3️⃣ Define the limits of integration
4️⃣ Set up the definite integral
5️⃣ Evaluate the integral
For rotation around the y-axis, the volume is calculated using the integral \( V = \int_{c}^{d} \pi [g(y)]^2 dy \)
True
What are the bounds of integration for the region bounded by \( y = x^2 \), \( y = 0 \), and \( x = 2 \) when rotated around the x-axis?
0 to 2
Match the axis of revolution with the correct variables and disk orientation:
x-axis ↔️ Variables: x, Orientation: Horizontal
y-axis ↔️ Variables: y, Orientation: Vertical
What is the axis of revolution when rotating the region bounded by \( y = \sqrt{x} \), \( x = 0 \), and \( x = 4 \) around the x-axis?
x-axis
Match the axis of rotation with the correct volume formula:
x-axis ↔️ ∫ₐ^ᵇ π [f(x)]^2 dx
y-axis ↔️ ∫ₒ^ᵈ π [g(y)]^2 dy
The axis of revolution can only be the x-axis.
False
Match the axis of revolution with the correct radius formula:
x-axis ↔️ \( f(x) \)
y-axis ↔️ \( g(y) \)
Steps to define the limits of integration
1️⃣ Determine the axis of revolution
2️⃣ Identify the starting and ending points
3️⃣ Set the integration limits
The volume of each disk in the disk method is given by πr²h
The axis of revolution is the line around which a region is rotated to form a solid
For rotation around the x-axis, the radius formula is \( f(x) \)
What is the first step in defining the limits of integration?
Determine the axis of revolution
To identify the starting and ending points, find the values of x or y where the region begins
What are the limits of integration for a region rotated around the y-axis?
c to d
In the example, the starting point is x = 0
Match the axis of revolution with the correct volume formula:
x-axis ↔️ V=∫abπ[f(x)]2dx
y-axis ↔️ V=∫cdπ[g(y)]2dy
For x-axis rotation, the integral form for volume is \( V = \int_{a}^{b} \pi [f(x)]^2 dx \).
True
What is the disk method used to find?
Volume of a solid of revolution
For rotation around the x-axis, the volume is calculated using the integral \( V = \int_{a}^{b} \pi [f(x)]^2 dx \)
True
For rotation around the y-axis, the radius is \( r = g(y) \) and the height is \( h = \)dy
When rotating the region bounded by \( y = x^2 \), \( y = 0 \), and \( x = 2 \) around the x-axis, the volume is \( V = \int_{0}^{2} \pi (x^2)^2 dx = \pi \int_{0}^{2} \)x^4
What is the radius when rotating the region bounded by \( x = y^2 \), \( y = 0 \), and \( y = 2 \) around the y-axis?
y^2
The disk method sums the volumes of infinitesimally thin disks to find the volume of a solid of revolution.
True
What is the axis of revolution when rotating the region bounded by \( y = \sqrt{x} \), \( x = 0 \), and \( x = 4 \) around the x-axis?
x-axis
When rotating around the x-axis, the radius formula is \( r = f(x) \), and the variable is x
If rotating around the x-axis, the radius is the value of the function f(x)
What are the limits of integration for rotation around the x-axis?
a to b
What is the formula for the volume when rotating around the x-axis using the disk method?
\( V = \int_{a}^{b} \pi [f(x)]^2 dx \)
What type of disk orientation is used when rotating around the x-axis?
Horizontal
Which variable is used when rotating around the y-axis?