8.13 Volume with Shell Method: Revolving Around the x- or y-Axis

    Cards (26)

    • What is the shell method used to calculate?
      Volume of solids
    • When revolving around the x-axis, the formula for the volume is V = 2π∫_a^b y f(y) dy
    • The shell method is useful when disk or washer methods are difficult to set up.
    • The first step in setting up the shell method integral is to identify the axis of revolution
    • The limits of integration in the shell method are determined based on the range of xx or yy.
    • What variable of integration is used when revolving around the x-axis in the shell method?
      yy
    • Arrange the key steps to set up the integral for the shell method in the correct order.
      1️⃣ Identify the axis of revolution
      2️⃣ Express functions in terms of the relevant variable
      3️⃣ Determine the limits of integration
    • The shell method is particularly useful when disks or washers are difficult to set up.
    • What is the expression for xx in terms of yy for the region bounded by y=y =x2 x^{2} and y=y =4 4?

      x=x =y \sqrt{y}
    • To find the volume of the solid obtained by revolving the region bounded by y=y =x2 x^{2} and y=y =4 4 about the y-axis, the limits of integration are from 00 to 22.
    • The volume of the solid obtained by revolving the region bounded by y=y =x2 x^{2} and y=y =4 4 about the y-axis is 8π8\pi
    • What is the formula for finding the volume when revolving around the y-axis using the shell method?
      V=V =2πabxf(x)dx 2\pi \int_{a}^{b} x f(x) \, dx
    • When revolving the region bounded by y=y =x2 x^{2} and y=y =4 4 about the y-axis, x</latex> in terms of yy is \sqrt{y}
    • What is the integral setup to find the volume when revolving the region bounded by y=y =x2 x^{2} and y=y =4 4 about the y-axis?

      V=V =2π04yydy 2\pi \int_{0}^{4} y \sqrt{y} \, dy
    • Identifying the axis of revolution is essential for setting up the correct integral in the shell method.
    • Match the axis of revolution with its corresponding shell orientation and variable of integration:
      x-axis ↔️ Vertical shells, yy
      y-axis ↔️ Horizontal shells, xx
    • What is the formula for finding the volume when revolving around the x-axis using the shell method?
      V=V =2πabyf(y)dy 2\pi \int_{a}^{b} y f(y) \, dy
    • When revolving the region bounded by y=y =x2 x^{2} and y=y =4 4 about the y-axis, the height of the shells is 4 - x^{2}
    • Revolving around the y-axis requires horizontal shells and integrating with respect to xx.
    • Steps to set up the integral for volume using the shell method:
      1️⃣ Identify the axis of revolution
      2️⃣ Express functions in terms of the correct variable
      3️⃣ Determine limits of integration
    • What is the formula for volume when revolving around the y-axis using the shell method?
      V=V =2πabxf(x)dx 2\pi \int_{a}^{b} x f(x) \, dx
    • What is the formula for volume when revolving around the x-axis using the shell method?
      V = 2\pi \int_{a}^{b} y f(y) \, dy</latex>
    • The axis of revolution determines the orientation of the cylindrical shells
    • Steps to evaluate the integral to find the volume using the shell method:
      1️⃣ Set up the integral
      2️⃣ Simplify the integrand
      3️⃣ Find the antiderivative
      4️⃣ Evaluate the antiderivative at the limits of integration
      5️⃣ Calculate the final volume
    • What is the antiderivative of 4xx34x - x^{3}?

      2x^{2} - \frac{x^{4}}{4}</latex>
    • What is the volume of the solid formed by revolving the region bounded by y=y =x2 x^{2} and y=y =4 4 around the y-axis?

      8π8\pi