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

• What is the washer method used to find?

  Volume of a solid
• When revolving around lines other than the x or y-axis, the radii must be adjusted by adding or subtracting the distance from the axis of revolution to the function.
• The formula for the volume using the washer method is $V =$$\pi \int_{a}^{b} [(R(x))^{2} - (r(x))^{2}] dx$
• What does $R(x)$ represent in the washer method formula?


  Outer radius
• What does $r(x)$ represent in the washer method formula?


  Inner radius
• Match the axis of revolution with the correct outer and inner radii formulas:

  $y =$$k$ ↔️ $R(x) =$$f(x) - k$ and $r(x) =$$g(x) - k$

  $x =$$k$ ↔️ $R(x) =$$k - f(y)$ and $r(x) =$$k - g(y)$
• When revolving around the line $y =$$k$, the outer radius is given by $f(x) - k$
• When revolving around the line $x =$$k$, the outer radius is given by $k - f(y)$
• What is the volume of the solid generated by revolving the region bounded by $f(x) =$$x^{2}$ andg(x) = x</latex> about the line $y =$$- 1$ from $x =$$0$ to $x =$$1$?


  $\pi \int_{0}^{1} [(x + 1)^{2} - (x^{2} +$$1)^{2}] dx$
• The volume of a solid using the washer method is calculated using an integral
• In the washer method, $R(x)$ represents the outer radius.
• In the washer method, $r(x)$ represents the inner radius.
• What is the outer radius when revolving the region bounded by $f(x) =$$x^{2}$ and $g(x) =$$x$ about the line $y =$$- 1$?


  $x +$$1$
• What is the inner radius when revolving the region bounded by $f(x) =$$x^{2}$ and $g(x) =$$x$ about the line $y =$$- 1$?


  $x^{2} +$$1$
• What does $R(x)$ represent in the washer method?


  Outer radius
• The radii in the washer method are adjusted based on the axis of revolution
• If the axis of revolution is $y =$$k$, the outer radius is $f(x) - k$ and the inner radius is $g(x) - k$
• What is the formula for the volume of a solid using the washer method?

  V = \pi \int_{a}^{b} [(R(x))^{2} - (r(x))^{2}] dx</latex>
• When revolving around $x =$$k$, the outer radius is $k - f(y)$ and the inner radius is $k - g(y)$
• Match the axis of revolution with the correct outer and inner radii:

  $y =$$k$ ↔️ $f(x) - k$ and $g(x) - k$

  $x =$$k$ ↔️ $k - f(y)$ and $k - g(y)$
• What are the radii if you revolve around $y =$$k$ and f(x) ≥ g(x)</latex>?


  $R(x) =$$f(x) - k$ and $r(x) =$$g(x) - k$
• The limits of integration in the washer method are the x-values where the bounding functions intersect
• Steps to determine the limits of integration:

  1️⃣ Set $f(x)$ equal to $g(x)$

  2️⃣ Solve for $x$

  3️⃣ Assign the smallest $x$-value as $a$ and the largest as $b$
• The limits of integration define the range over which we calculate the volume.
• The washer method is used to calculate the volume of a solid formed by revolving a region around an axis
• What is the general formula for the volume using the washer method?

  $V =$$\pi \int_{a}^{b} [(R(x))^{2} - (r(x))^{2}] dx$
• In the washer method, $R(x)$ represents the outer radius.
• In the washer method, $r(x)$ represents the inner radius
• Match the axis of revolution with its corresponding outer and inner radii adjustments:

  $y =$$k$ ↔️ $f(x) - k$ and $g(x) - k$

  $x =$$k$ ↔️ $k - f(y)$ and $k - g(y)$
• The washer method is particularly useful when the solid has a hole
• The inner radius $r(x)$ must always be greater than the outer radius $R(x)$.


  False
• Steps to determine the inner and outer radii in the washer method:

  1️⃣ Identify the bounding functions

  2️⃣ Determine the axis of revolution

  3️⃣ Adjust the functions to reflect distances from the axis
• To identify the limits of integration, set the bounding functions equal to each other
• What is the final step in calculating the volume using the washer method?

  Evaluate the integral