8.12 Volume with Washer Method: Revolving Around Other Horizontal or Vertical Lines

• The outer radius in the washer method is the distance from the axis to the outer boundary.
• The disk method is used when the area being revolved touches the axis of revolution.

  True
• Steps to identify the region to be revolved

  1️⃣ Sketch the functions on a coordinate plane

  2️⃣ Determine the intersection points

  3️⃣ Shade the area between the functions
• Does the area touch the axis of revolution in the washer method?

  No
• Steps to determine the axis of rotation

  1️⃣ Identify the given information

  2️⃣ Consider the orientation of the region

  3️⃣ Use symmetry if applicable
• What is the formula for calculating volume using the washer method?

  $V =$$\pi \int_{a}^{b} [R(x)^{2} - r(x)^{2}] \, dx$
• The washer method is used when the area being revolved creates a hollow solid.

  True
• Does the area touch the axis in the disk method?

  Yes
• What is the volume formula for the disk method?

  $V =$$\pi \int_{a}^{b} R(x)^{2} \, dx$
• If the region is horizontal, the axis of rotation is vertical
• Match the axis with the variable and formula for the outer radius:

  Horizontal Axis ↔️ $x$, $R(x) =$$(\text{Axis value}) - (\text{Function value})$

  Vertical Axis ↔️ $y$, $R(y) =$$(\text{Axis value}) - (\text{Function value})$
• The inner radius formula for a horizontal axis is $r(x) =$$|(\text{Axis value}) - (\text{Function value})|$.


  True
• Comparing inner and outer radii:

  1️⃣ Inner radius is closer to the axis

  2️⃣ Outer radius is further from the axis

  3️⃣ The formula for the inner radius is $r(x) =$$|y_{\text{axis}} - y_{\text{inner}}|$

  4️⃣ The formula for the outer radius is $R(x) =$$|y_{\text{axis}} - y_{\text{outer}}|$
• The volume in the washer method is calculated using the formula V = \pi \int_{a}^{b} [R(x)^{2} - r(x)^{2}] \, dx</latex>, where $R(x)$ is the outer radius.
• The inner radius $r(x)$ in the washer method represents the distance from the axis to the inner boundary
• What do the intersection points of the functions define in the washer method?

  Limits of integration
• What is the washer method used to calculate?

  Volume of a hollow solid
• What is the formula for calculating volume using the washer method?

  V = \pi \int_{a}^{b} [R(x)^{2} - r(x)^{2}] \, dx</latex>
• The washer method requires two types of radii: outer and inner.
• The washer method creates a solid with a hollow center when revolved around an axis.


  True
• To visualize the region to be revolved, you should first sketch the functions on a coordinate plane.
• What type of radii are used in the washer method?

  Outer and inner
• In the washer method, the area being revolved does not touch the axis
• The volume in the washer method is found by integrating the difference of the squared radii
• The washer method uses two radii: outer and inner.


  True
• Steps to identify the region to be revolved:

  1️⃣ Sketch the functions on a coordinate plane

  2️⃣ Determine the intersection points

  3️⃣ Shade the area between the functions
• What is the first step in determining the axis of rotation?

  Identify given information
• If a region bounded by $y =$$x^{2}$ and $y =$$4$ is revolved around y = 6</latex>, the axis of rotation is $y =$$6$.


  True
• Revolving $y =$$x^{2}$ around $y =$$4$ gives an outer radius of R(x) = 4 - x^{2}</latex>.True
• What is the inner radius if $y =$$x^{2}$ is revolved around $y =$$4$?


  $r(x) =$$|4 - x^{2}|$
• What is the key difference between the disk and washer methods in terms of the area touching the axis?

  Disk: Yes, Washer: No
• What does the outer radius $R(x)$ represent in the washer method?


  Distance from axis to outer boundary
• To visualize the region being revolved, it is necessary to sketch the functions
• If the region is horizontal, the axis of rotation is typically a vertical
• What is the outer radius when revolving around $y =$$6$ with $y =$$x^{2}$ and $y =$$4$ as boundaries?


  $R(x) =$$6 - x^{2}$
• The formula for calculating the volume of a solid of revolution around a vertical axis using the washer method is $V =$$\pi \int_{a}^{b} [R(y)^{2} - r(y)^{2}] \, dy$
• The intersection points of $y =$$x^{2}$ and $y =$$4$ are $x =$$\pm 2$
• The integral for the volume of the region bounded by y = x^{2}</latex> and $y =$$4$ revolved around $y =$$6$ is $V =$$\pi \int_{ - 2}^{2} [(6 - x^{2})^{2} - 2^{2}] \, dx$
• The volume integral for the region bounded by $y =$$x^{2}$ and $y =$$4$ revolved around $y =$$6$ is $V =$$\pi \int_{ - 2}^{2} [(6 - x^{2})^{2} - 2^{2}] \, dx$
• What is the general formula for the volume of a solid of revolution around a vertical axis using the washer method?

  $V =$$\pi \int_{c}^{d} [R(y)^{2} - r(y)^{2}] \, dy$