Setting up integrals using the washer method:

• What is the washer method used for?

  Finding the volume of solids with a hole
• Steps to set up an integral using the washer method

  1️⃣ Identify the generating curve

  2️⃣ Visualize the solid as a stack of washers

  3️⃣ Determine the radius of each washer

  4️⃣ Set up the integral
• The washer method uses the integral formula \pi
• When the axis of revolution is the y-axis, the washers are oriented vertically
• Match the axis of revolution with its corresponding washer orientation and integral form:

  x-axis ↔️ Horizontal washers, \(\pi \int_{a}^{b} [R(x)^{2} - r(x)^{2}] \, dx\)

  y-axis ↔️ Vertical washers, \(\pi \int_{c}^{d} [R(y)^{2} - r(y)^{2}] \, dy\)
• Match the axis of revolution with its corresponding outer and inner radii:

  x-axis ↔️ R(x), r(x)

  y-axis ↔️ R(y), r(y)
• Steps to set up the volume integral using the washer method:

  1️⃣ Sketch the region

  2️⃣ Identify the axis of revolution

  3️⃣ Determine outer and inner radii

  4️⃣ Set up the integral
• Match the axis of revolution with its corresponding washer orientation and integral form:

  x-axis ↔️ Horizontal washers, \(\pi \int_{a}^{b} [R(x)^{2} - r(x)^{2}] \, dx\)

  y-axis ↔️ Vertical washers, \(\pi \int_{c}^{d} [R(y)^{2} - r(y)^{2}] \, dy\)
• Expressing the radii as functions of the independent variable is crucial for the washer method.


  True
• Match the axis of revolution with the correct outer and inner radii:

  x-axis ↔️ R(x), r(x)

  y-axis ↔️ R(y), r(y)
• The choice of axis of revolution (x-axis or y-axis) determines how the radii are expressed
• What is the inner radius in the washer method?

  Distance to inner edge
• The radii must be expressed as functions of the independent variable for the washer method.


  True
• The limits of integration are found by determining the intersection points of the generating curves.


  True
• What type of functions are used for the radii when the axis of revolution is the x-axis?

  Functions of x
• When the axis of revolution is the y-axis, the washers stack vertically
• In the washer method, the outer radius is denoted by R(x)
• Match the axis of revolution with the correct integral setup:

  x-axis ↔️ $V =$$\pi \int_{a}^{b} [R(x)^{2} - r(x)^{2}] \, dx$

  y-axis ↔️ $V =$$\pi \int_{c}^{d} [R(y)^{2} - r(y)^{2}] \, dy$
• Identifying the axis of revolution is crucial for setting up the washer method integral.


  True
• The axis of revolution influences the choice of radii and integration variable
• The outer radius represents the distance from the axis of revolution to the outer edge of the washer
• The limits of integration are determined by the points where the generating curves intersect
• The washer method formula for rotation about the x-axis is \(\pi \int_{a}^{b} [R(x)^{2} - r(x)^{2}] \, dx\)
• What does R(x) or R(y) represent in the washer method?

  Outer radius
• What does r(x) or r(y) represent in the washer method?

  Inner radius
• Expressing the radii as functions of the independent variable is essential for the washer method.


  True
• Match the axis of revolution with the correct outer and inner radii:

  x-axis ↔️ R(x), r(x)

  y-axis ↔️ R(y), r(y)
• The axis of revolution determines how the radii are expressed
• What do the limits of integration define in the washer method?

  Boundaries of the solid
• What are the limits of integration for the y-axis?

  Intersection y-coordinates
• The outer radius R(x) or R(y) represents the distance from the axis of revolution to the outer curve.


  True
• The integral formula for rotation about the x-axis is $V =$$\pi \int_{a}^{b} [R(x)^{2} - r(x)^{2}] \, dx$


  True
• The formula for the volume using the washer method is V = \pi \int_{a}^{b} [R(x)^2 - r(x)^2] \, dx</latex>


  True
• What does \(R(x)\) represent in the washer method?

  Outer radius
• What type of washers are used when the axis of revolution is the x-axis?

  Horizontal
• When rotating about the x-axis, the washers stack horizontally and the radii are functions of x.

  True
• The inner radius is denoted as r(x) when rotating about the y-axis.

  False
• When rotating about the y-axis, the limits of integration are the y-coordinates of the intersection points.


  True
• The washer method can only be used when there is a hole in the solid of revolution.


  True
• The choice of axis of revolution (x-axis or y-axis) determines how the radii are expressed