Setting up integrals using the disk method:

• What is the disk method used to find?

  Volume of a solid of revolution
• When rotating around the y-axis, the radius is expressed as a function of x.

  False
• If the axis of rotation is the x-axis, the integral is set up as \(\pi \int_{a}^{b} [r(x)]^2 dx\), where r(x) represents the radius function.
• Steps to use the disk method

  1️⃣ Identify the region to be revolved

  2️⃣ Determine the radius function r(x) or r(y)

  3️⃣ Define the limits of integration

  4️⃣ Set up and evaluate the integral
• Steps to set up the disk method integral

  1️⃣ Choose the axis of rotation

  2️⃣ Determine the radius function

  3️⃣ Set up the integral
• The first step in the disk method is to identify the radius
• The integral setup for the disk method depends on the axis of rotation


  True
• Match the axis of rotation with its integral setup:

  x-axis ↔️ \(\pi \int_{a}^{b} [r(x)]^2 dx\)

  y-axis ↔️ \(\pi \int_{c}^{d} [r(y)]^2 dy\)
• The radius function for the disk method depends on the choice of axis
• If rotating around the x-axis, the radius function is \(r(x)\)
• When rotating around the x-axis, the integral setup is \(\pi \int_{a}^{b} [r(x)]^2 dx\)
• The limits of integration in the definite integral for volume are determined by the region being revolved.

  True
• When rotating around the y-axis, the integral is set up as \(\pi \int_{a}^{b} [r(x)]^2 dx\).

  False
• The axis of revolution determines the variable with respect to which we integrate.
• When rotating around the x-axis, the integral setup is \(\pi \int_{a}^{b} [r(x)]^2 dx\)
• The axis of revolution in the disk method determines the variable with respect to which we integrate


  True
• What variable do you integrate with respect to when rotating around the y-axis?

  y
• What are the limits of integration when rotating around the x-axis?

  a to b
• What is the first step in expressing the radius function for the disk method?

  Identify the region
• The first step in expressing the radius function for the disk method is to identify the region to be revolved.


  True
• Match the axis of rotation with the corresponding integral setup:

  x-axis ↔️ \(\pi \int_{a}^{b} [r(x)]^2 dx\)

  y-axis ↔️ \(\pi \int_{c}^{d} [r(y)]^2 dy\)
• When rotating around the x-axis, the radius function is \(r(x)\)
• Match the axis of rotation with its corresponding integral setup:

  x-axis ↔️ \(\pi \int_{a}^{b} [r(x)]^2 dx\)

  y-axis ↔️ \(\pi \int_{c}^{d} [r(y)]^2 dy\)
• To use the disk method, you first need to identify the region that will be revolved around an axis.
• What does the radius function describe in the disk method?

  Distance from axis of rotation
• When rotating around the x-axis, the limits of integration are along the y-axis.

  False
• The radius function describes the distance from the axis of rotation to the boundary
• The disk method is used to find the volume of a solid of revolution


  True
• Steps to use the disk method

  1️⃣ Identify the radius function

  2️⃣ Choose the axis of rotation

  3️⃣ Set up the integral
• What must you identify first when using the disk method?

  The region
• The radius function describes the distance from the axis of rotation to the boundary
• When rotating around the x-axis, the radius function must be in terms of x


  True
• If rotating the region bounded by $y =$$x^{2}$ from $x =$$0$ to $x =$$2$ around the x-axis, the integral is \(\pi \int_{0}^{2} (x^2)^2 dx\)
• When rotating around the y-axis, the radius function is r(y)</latex>, which describes the distance from the y-axis to the boundary of the region.

  True
• If revolving around the x-axis, the integral setup for the volume is \(\pi \int_{a}^{b} [r(x)]^{2} dx\).


  True