Cards (57)

    • The volume of each disk in the disk method is given by \( V = \pi r^2 h \), where \( r \) is the radius and \( h \) is the thickness
    • For rotation around the x-axis, the radius is \( r = f(x) \) and the height is \( h = \)dx
    • Match the axis of rotation with the corresponding variables and height:
      x-axis ↔️ Variables: x, Height: dx
      y-axis ↔️ Variables: y, Height: dy
    • What is the axis of revolution when a region is rotated to form a solid of revolution?
      Line of rotation
    • The volume when rotating the region bounded by \( x = y^2 \), \( y = 0 \), and \( y = 2 \) around the y-axis is \( V = \int_{0}^{2} \pi (y^2)^2 dy = \pi \int_{0}^{2} \)y^4
    • For rotation around the x-axis, the volume is calculated using the integral \( V = \int_{a}^{b} \pi [f(x)]^2 dx \), where \( f(x) \) represents the radius
    • The volume of the solid formed by rotating the region bounded by \( y = \sqrt{x} \), \( x = 0 \), and \( x = 4 \) around the x-axis is \( V = \int_{0}^{4} \pi (\sqrt{x})^2 dx \).
      True
    • When rotating around the y-axis, the radius for \( x = y^2 \) is \( r = y^2 \).
      True
    • For the region bounded by \( y = x^2 \), \( y = 0 \), and \( x = 2 \), the limits of integration for rotation around the x-axis are from 0 to 2.

      True
    • Steps to set up integrals using the disk method
      1️⃣ Identify the axis of revolution
      2️⃣ Determine the radius of each disk
      3️⃣ Define the limits of integration
      4️⃣ Set up the definite integral
      5️⃣ Evaluate the integral
    • For rotation around the y-axis, the volume is calculated using the integral \( V = \int_{c}^{d} \pi [g(y)]^2 dy \)

      True
    • What are the bounds of integration for the region bounded by \( y = x^2 \), \( y = 0 \), and \( x = 2 \) when rotated around the x-axis?
      0 to 2
    • Match the axis of revolution with the correct variables and disk orientation:
      x-axis ↔️ Variables: x, Orientation: Horizontal
      y-axis ↔️ Variables: y, Orientation: Vertical
    • What is the axis of revolution when rotating the region bounded by \( y = \sqrt{x} \), \( x = 0 \), and \( x = 4 \) around the x-axis?
      x-axis
    • Match the axis of rotation with the correct volume formula:
      x-axis ↔️ ∫ₐ^ᵇ π [f(x)]^2 dx
      y-axis ↔️ ∫ₒ^ᵈ π [g(y)]^2 dy
    • The axis of revolution can only be the x-axis.
      False
    • Match the axis of revolution with the correct radius formula:
      x-axis ↔️ \( f(x) \)
      y-axis ↔️ \( g(y) \)
    • Steps to define the limits of integration
      1️⃣ Determine the axis of revolution
      2️⃣ Identify the starting and ending points
      3️⃣ Set the integration limits
    • The volume of each disk in the disk method is given by πr²h
    • The axis of revolution is the line around which a region is rotated to form a solid
    • For rotation around the x-axis, the radius formula is \( f(x) \)
    • What is the first step in defining the limits of integration?
      Determine the axis of revolution
    • To identify the starting and ending points, find the values of x or y where the region begins
    • What are the limits of integration for a region rotated around the y-axis?
      c to d
    • In the example, the starting point is x = 0
    • Match the axis of revolution with the correct volume formula:
      x-axis ↔️ V=V =abπ[f(x)]2dx \int_{a}^{b} \pi [f(x)]^{2} dx
      y-axis ↔️ V=V =cdπ[g(y)]2dy \int_{c}^{d} \pi [g(y)]^{2} dy
    • For x-axis rotation, the integral form for volume is \( V = \int_{a}^{b} \pi [f(x)]^2 dx \).

      True
    • What is the disk method used to find?
      Volume of a solid of revolution
    • For rotation around the x-axis, the volume is calculated using the integral \( V = \int_{a}^{b} \pi [f(x)]^2 dx \)
      True
    • For rotation around the y-axis, the radius is \( r = g(y) \) and the height is \( h = \)dy
    • When rotating the region bounded by \( y = x^2 \), \( y = 0 \), and \( x = 2 \) around the x-axis, the volume is \( V = \int_{0}^{2} \pi (x^2)^2 dx = \pi \int_{0}^{2} \)x^4
    • What is the radius when rotating the region bounded by \( x = y^2 \), \( y = 0 \), and \( y = 2 \) around the y-axis?
      y^2
    • The disk method sums the volumes of infinitesimally thin disks to find the volume of a solid of revolution.

      True
    • What is the axis of revolution when rotating the region bounded by \( y = \sqrt{x} \), \( x = 0 \), and \( x = 4 \) around the x-axis?
      x-axis
    • When rotating around the x-axis, the radius formula is \( r = f(x) \), and the variable is x
    • If rotating around the x-axis, the radius is the value of the function f(x)
    • What are the limits of integration for rotation around the x-axis?
      a to b
    • What is the formula for the volume when rotating around the x-axis using the disk method?
      \( V = \int_{a}^{b} \pi [f(x)]^2 dx \)
    • What type of disk orientation is used when rotating around the x-axis?
      Horizontal
    • Which variable is used when rotating around the y-axis?
      y
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