8.13 Volume with Shell Method: Revolving Around the x- or y-Axis

• The Shell Method is used to find the volume of a 3D object formed by revolving a 2D region
• The Disk Method is used when the 2D region is revolved around a parallel axis, while the Shell Method is used when the region is revolved around a perpendicular axis.

  False
• In the Shell Method, the volume formula uses the radius of the shell, which is x
• What is the primary difference between the Shell Method and the Disk Method for calculating volume?

  Axis of revolution
• The axis of revolution in the Shell Method can only be the x-axis.

  False
• When revolving around the y-axis, the integral is taken with respect to x.

  False
• Steps to identify the region to be rotated in the Shell Method:

  1️⃣ Understand the bounded area

  2️⃣ Sketch the graph

  3️⃣ Determine boundaries
• Match the shape of the region with its corresponding functions:

  Rectangle ↔️ f(x) = c, g(x) = d, x = a, x = b

  Triangle ↔️ f(x) = mx + c, x = a, x = b

  Curved Area ↔️ f(x), g(x), x = a, x = b
• Steps in the Shell Method

  1️⃣ Identify the 2D region and axis of revolution

  2️⃣ Set up an integral expression for the volume

  3️⃣ Evaluate the integral
• Match the axis of revolution with the correct volume formula:

  x-axis ↔️ V = 2\pi \int_{a}^{b} f(x)x\,dx</latex>

  y-axis ↔️ $V =$$2\pi \int_{c}^{d} g(y)y\,dy$
• If the region is revolved around the y-axis, the integral is taken with respect to y, and g(y) represents the width of the shell.

  True
• The Shell Method is used to calculate the volume of a 3D object created by revolving a 2D region around an axis
• Match the axis of revolution with its corresponding formula in the Shell Method:

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

  y-axis ↔️ $V =$$2\pi \int_{c}^{d} g(y)y\,dy$
• The region to be rotated in the Shell Method is enclosed by given functions or equations
• Different shapes of regions in the Shell Method include triangles and curved areas.


  True
• When using the Shell Method, the radius and height depend on the axis of revolution
• When revolving around the x-axis, the height function is f(x)
• Match the axis of revolution with its corresponding volume formula:

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

  y-axis ↔️ $V =$$2\pi \int_{c}^{d} g(y)y \, dy$
• When revolving around the x-axis, the volume formula is V = 2\pi \int_{a}^{b} f(x)x \, dx
• Match the axis of revolution with its corresponding radius and height functions:

  x-axis ↔️ Radius: x, Height: f(x)

  y-axis ↔️ Radius: y, Height: g(y)
• For x-axis revolution, the radius function is x.


  True
• For x-axis revolution, the height function is f(x)
• The final step in using the Shell Method is to set up the integral
• The Shell Method is an alternative to the Disk Method.

  True
• The axis of revolution can be either the x-axis or the y-axis
• What is the integration variable when revolving around the y-axis using the Shell Method?

  y
• Sketching the graph helps visualize the region to be rotated.

  True
• Match the region shape with its corresponding functions:

  Rectangle ↔️ f(x) = c, g(x) = d, x = a, x = b

  Triangle ↔️ f(x) = mx + c, x = a, x = b

  Curved Area ↔️ f(x), g(x), x = a, x = b
• When revolving around the x-axis, the radius function is x and the height function is f(x).


  True
• What is the volume formula when revolving around the x-axis using the Shell Method?

  $V =$$2\pi \int_{a}^{b} f(x)x \, dx$
• For an x-axis revolution, the radius function is always x.


  True
• Steps to set up the Shell Method integral for a solid of revolution

  1️⃣ Identify Axis of Revolution

  2️⃣ Express Radius Function

  3️⃣ Express Height Function

  4️⃣ Determine Limits of Integration

  5️⃣ Set up the Integral
• What is the final step in setting up the Shell Method integral?

  Set up the Integral
• The integral V = 2\pi \int_{0}^{2} (4 - x^{2})x \, dx</latex> represents the volume when revolving around the y-axis.

  False
• When revolving around the x-axis, the radius of each shell is x.


  True
• Steps to set up the integral for the Shell Method:

  1️⃣ Identify axis of revolution

  2️⃣ Express radius function

  3️⃣ Express height function

  4️⃣ Determine limits of integration

  5️⃣ Set up the integral
• What does the appropriate formula for calculating volume using the Shell Method depend on?

  Axis of revolution
• What is the volume formula when revolving around the y-axis using the Shell Method?

  V = 2\pi \int_{c}^{d} g(y)y \, dy</latex>
• Steps to calculate volume using the Shell Method:

  1️⃣ Identify the axis of revolution

  2️⃣ Express the radius function

  3️⃣ Express the height function

  4️⃣ Determine the limits of integration

  5️⃣ Set up the integral
• What is the radius function when revolving around the y-axis using the Shell Method?

  y