8.9 Volume with Disc Method: Revolving Around the x- or y-Axis

• The disc method calculates the volume of a solid of revolution by slicing it into thin discs
• Match the axis of revolution with the correct volume formula:

  x-axis ↔️ ∫ π [f(x)]² dx

  y-axis ↔️ ∫ π [g(y)]² dy
• What is the disc radius when revolving around the y-axis?

  g(y)
• The disc method requires slicing the solid into discs perpendicular to the axis of revolution.


  True
• What is the volume of the solid formed by revolving f(x) = √x around the x-axis from x = 0 to x = 4?

  8π
• The formula for revolving around the x-axis is V = ∫ π [f(x)]² dx.

  True
• When revolving around the y-axis, the disc radius is given by g(y).

  True
• What is the volume of the solid formed by revolving g(y) = √y around the y-axis from y = 0 to y = 9?

  81π/2
• What is the volume formula for revolving around the y-axis?

  ∫ π [g(y)]² dy
• The general formula for calculating volume using the disc method when revolving around the y-axis is ∫ π [g(y)]² dy
• Match the axis of revolution with the correct volume formula:

  x-axis ↔️ ∫ π [f(x)]² dx

  y-axis ↔️ ∫ π [g(y)]² dy
• The disc method uses integration to sum the volumes of thin discs to find the total volume of a solid.


  True
• What is the disc radius when revolving around the x-axis?

  f(x)
• The volume of each disc is calculated as πr²h, where r is the radius
• Steps to calculate the volume of a solid formed by revolving f(x) = √x around the x-axis from x = 0 to x = 4

  1️⃣ Square the function: [f(x)]² = x

  2️⃣ Multiply by π: π [f(x)]² = πx

  3️⃣ Integrate over the interval: ∫ πx dx

  4️⃣ Evaluate the definite integral: 8π
• The volume of each disc is calculated as πr²h, where h is the thickness
• Steps to calculate the volume of a solid formed by revolving f(x) = √x around the x-axis from x = 0 to x = 4

  1️⃣ Square the function: [f(x)]² = x

  2️⃣ Multiply by π: π [f(x)]² = πx

  3️⃣ Integrate over the interval: ∫ πx dx

  4️⃣ Evaluate the definite integral: 8π
• Revolving f(x) = √x around the x-axis from x = 0 to x = 4 yields a volume of 8π
• What is the formula for revolving around the y-axis using the disc method?

  V = ∫ π [g(y)]² dy
• Steps to calculate the volume of a solid formed by revolving g(y) = √y around the y-axis from y = 0 to y = 9

  1️⃣ Square the function: [g(y)]² = y

  2️⃣ Multiply by π: π [g(y)]² = πy

  3️⃣ Integrate over the interval: ∫ πy dy

  4️⃣ Evaluate the definite integral: 81π/2
• When revolving around the x-axis, the volume formula is ∫ π [f(x)]² dx
• What is represented by 'V' in the volume formula for the disc method?

  Volume
• When revolving around the x-axis, the volume formula uses the function f(x).


  True
• The volume of each disc in the disc method is calculated using the formula πr²h
• What is the volume of the solid formed by revolving f(x) = √x around the x-axis from x = 0 to x = 4?

  8π
• What is the final volume of the solid formed by revolving f(x) = √x around the x-axis from x = 0 to x = 4?

  8π
• Revolving g(y) = √y around the y-axis from y = 0 to y = 9 results in a volume of 81π/2 cubic units.


  True
• The integral setup for the x-axis rotation involves squaring the function and multiplying by π
• The volume formula for x-axis rotation is V = ∫ π [f(x)]² dx
• What is the squared value of f(x) = x²?

  $x^{4}$
• To calculate the volume of a solid formed by revolving a region around the x-axis, we multiply by π
• In the volume formula for x-axis rotation, the function is first squared
• What is the key difference between x-axis and y-axis rotation in volume calculation?

  Function and variable
• The volume formula for y-axis rotation uses g(y) as the function.


  True
• What is the squared value of f(x) = √x?

  x
• The volume of the solid is 8π cubic units.
• For f(x) = √x, the square of the function is x.
• When revolving f(x) = √x around the x-axis, the expression π [f(x)]² simplifies to πx.
• Steps to calculate the volume of a solid formed by revolving f(x) = √x around the x-axis from x = 0 to x = 4

  1️⃣ [f(x)]² = x

  2️⃣ π [f(x)]² = πx

  3️⃣ ∫ from 0 to 4 (πx) dx = 8π
• For g(y) = √y, the square of the function is y.