Setting up integrals using the shell method:

Cards (22)

  • What does the radius function r(x)r(x) or r(y)r(y) represent in the shell method?

    Distance from axis of revolution
  • What is the shell method used for in calculus?
    Finding volume of a solid
  • Steps for finding volume using the shell method
    1️⃣ Visualize the solid as cylindrical shells
    2️⃣ Set up an integral with the differential element
    3️⃣ Integrate to find the total volume
  • The axis of revolution determines the orientation of the **cylindricalshells
  • Match the visualization method with its differential element:
    Shell Method ↔️ Volume of a shell
    Disc Method ↔️ Volume of a disc
  • The shell method is particularly useful for complex solids where the disc method is difficult to apply.

    True
  • The axis of revolution defines the integration variable based on the shell orientation
  • In the shell method, the solid is visualized as a series of **cylindricalshells
  • For what type of solids is the shell method particularly useful?
    Irregularly-shaped solids
  • The axis of revolution must be identified before using the shell method.

    True
  • If the axis of revolution is the y-axis, what is r(x)r(x) for a point (x,y)(x, y) on the region?

    r(x)=r(x) =x x
  • The shell method is often preferred over the disc method for complex solids.

    True
  • Steps to calculate the volume of a solid of revolution using the shell method
    1️⃣ Visualize cylindrical shells around the axis of rotation
    2️⃣ Create an integral representing the volume of an individual shell
    3️⃣ Integrate to find the total volume
  • The axis of revolution determines the orientation of the cylindrical shells
  • The radius function represents the distance from the axis of revolution to a cylindrical shell
  • What is the radius function if the axis of revolution is the y-axis and a point on the region has coordinates (x,y)(x, y)?

    r(x)=r(x) =x x
  • The shell method formula for horizontal shells is V=V =∫cd2πr(y)h(y) dy \int_{c}^{d} 2\pi r(y)h(y) \, dy, where r(y)r(y) and h(y)h(y) depend on the geometry of the solid
  • Steps to set up the volume integral using the shell method
    1️⃣ Identify the axis of revolution
    2️⃣ Determine the radius function
    3️⃣ Determine the height function
    4️⃣ Apply the appropriate formula for vertical or horizontal shells
  • The shell method calculates the volume of a solid of revolution by visualizing it as stacked cylindrical shells
  • The height function in the shell method is determined by the difference between the upper and lower boundaries of the region being revolved.

    True
  • For vertical shells, the height function is h(x)=h(x) = f_{\text{upper}}(x) - f_{\text{\lower}}(x), while for horizontal shells, it is h(y)=h(y) = g_{\text{\right}}(y) - g_{\text{\left}}(y) height
  • What is the shell method formula for vertical shells?
    V = \int_{a}^{b} 2\pi r(x)h(x) \, dx</latex>