Cards (21)

  • When finding the volume of a solid, it is essential to identify the region of integration
  • If the cross sections are perpendicular to the x-axis, what is the region of integration?
    Between two x-values
  • What is the area expression for a square cross section perpendicular to the y-axis?
    A(x)
  • The area of a generic cross section depends on the axis of symmetry
  • When the axis of symmetry is horizontal, the area of the cross section is expressed as A(y)
  • What is the first step to set up an integral for the volume of a solid with known cross sections?
    Identify the axis of symmetry
  • What shape is a cross section likely to be if it is perpendicular to the y-axis?
    Square or triangle
  • The region of integration depends on the axis of symmetry of the solid.

    True
  • The shape of the cross sections depends on the axis of symmetry of the solid.

    True
  • The shape of the cross sections is crucial in determining the area expression for volume calculation.

    True
  • What is the area expression for cross sections perpendicular to the x-axis?
    A(y)
  • What shape might a cross section be if the axis of symmetry is vertical and parallel to the y-axis?
    Square or triangle
  • Steps to set up the volume integral for a solid with known cross sections
    1️⃣ Identify the axis of symmetry
    2️⃣ Determine the shape of the cross sections
    3️⃣ Set up the volume integral using A(x) or A(y)
  • If the axis of symmetry is vertical, the integration limits are the y-values where the region starts and ends.
    False
  • Steps to find the volume of a solid with known cross sections
    1️⃣ Identify the region of integration
    2️⃣ Determine the shape of the cross sections
    3️⃣ Find the area of a generic cross section in terms of xx or yy
    4️⃣ Set up the integral based on the cross section shape and its area
    5️⃣ Determine the integration limits
  • Match the axis of symmetry with the region of integration and shape of the cross section:
    Horizontal (parallel to the x-axis) ↔️ Between two y-values ||| Rectangle or any shape
    Vertical (parallel to the y-axis) ↔️ Between two x-values ||| Square or triangle
  • If the cross sections are squares perpendicular to the x-axis, the region of integration is between the x-values that bound the base
  • Match the axis of symmetry with the shape of the cross section and area expression:
    Horizontal (parallel to x-axis) ↔️ Rectangle or any shape ||| A(y)
    Vertical (parallel to y-axis) ↔️ Square or triangle ||| A(x)
  • What is the area expression for a cross section if the axis of symmetry is horizontal and parallel to the x-axis?
    A(y)
  • If the cross sections are perpendicular to the x-axis, the area of each cross section is a function of y, i.e., A(y).
    True
  • If the axis of symmetry is horizontal, the volume integral is \int_{y_{1}}^{y_{2}} A(y) dy