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

• What is the Disc Method used to calculate?

  Volume of a solid
• When revolving around the x-axis, the volume formula is $V =$$\pi \int_{a}^{b} [f(x)]^{2} dx$.
• For $y =$$x^{2}$ revolved around the x-axis from 0 to 1, the volume is $\pi \int_{0}^{1} (x^{2})^{2} dx =$$\pi \int_{0}^{1} x^{4} dx$, which simplifies to \pi/5
• Steps to find the volume when $y =$$x^{2}$ is revolved around the x-axis from 0 to 1


  1️⃣ Square the function: $[f(x)]^{2} =$$(x^{2})^{2} =$$x^{4}$

  2️⃣ Set up the integral: $V =$$\pi \int_{0}^{1} x^{4} dx$

  3️⃣ Evaluate the integral: $\pi \left[ \frac{x^{5}}{5} \right]_{0}^{1} =$$\pi \left( \frac{1}{5} - 0 \right)$

  4️⃣ Final Volume: $V =$$\frac{\pi}{5}$
• When revolving around the y-axis, the volume formula is V = \pi \int_{c}^{d} [g(y)]^{2} dy</latex>.
• For $x =$$\sqrt{y}$ revolved around the y-axis from 0 to 4, the volume is $\pi \int_{0}^{4} y dy$, which simplifies to 8\pi
• Steps to find the volume when $x =$$\sqrt{y}$ is revolved around the y-axis from 0 to 4


  1️⃣ Square the function: $[g(y)]^{2} =$$(\sqrt{y})^{2} =$$y$

  2️⃣ Set up the integral: $V =$$\pi \int_{0}^{4} y dy$

  3️⃣ Evaluate the integral: $\pi \left[ \frac{y^{2}}{2} \right]_{0}^{4} =$$\pi \left( \frac{16}{2} - 0 \right)$

  4️⃣ Final Volume: $V =$$8\pi$
• When revolving around the x-axis, the volume formula is V = \pi \int_{a}^{b} [f(x)]^{2} dx</latex>.
• When a region under a curve $y =$$f(x)$ is revolved around the x-axis, the volume $V$ can be found using the formula $V =$$\pi \int_{a}^{b} [f(x)]^{2} dx$, where $f(x)$ is the function being revolved
• What is the final volume wheny = x^{2}</latex> is revolved around the x-axis from 0 to 2?

  $V =$$\frac{32\pi}{5}$
• What happens when a region under a curve $y =$$f(x)$ is revolved around the x-axis?


  It forms a solid
• The formula for the volume when revolving around the x-axis is V = \pi \int_{a}^{b} [f(x)]^{2} dx
• What are the limits of integration when revolving $y =$$x^{2}$ around the x-axis from 0 to 2?


  0 to 2
• When revolving $y =$$x^{2}$ around the x-axis, the squared function is $(x^{2})^{2} =$$x^{4}$
• The formula for the volume when revolving around the y-axis is V = \pi \int_{c}^{d} [g(y)]^{2} dy
• What is the function in terms of y when $x =$$\sqrt{y}$ is revolved around the y-axis?


  $g(y) =$$\sqrt{y}$
• When revolving $x =$$\sqrt{y}$ around the y-axis, the squared function is $(\sqrt{y})^{2} =$$y$
• What is the final volume when $x =$$\sqrt{y}$ is revolved around the y-axis from 0 to 4?


  V = 8\pi</latex>
• Steps to calculate the volume of a solid formed by revolving around the x-axis

  1️⃣ Square the function $[f(x)]^{2}$

  2️⃣ Set up the integral $V =$$\pi \int_{a}^{b} [f(x)]^{2} dx$

  3️⃣ Evaluate the integral $\pi \left[ F(x) \right]_{a}^{b} =$$\pi [F(b) - F(a)]$

  4️⃣ Final Volume $V$
• The volume of the solid formed when y = x^{2}</latex> is revolved around the x-axis from 0 to 2 is $V =$$\frac{32\pi}{5}$
• What is the function squared when $y =$$\sqrt{x}$ is revolved around the x-axis from 0 to 4?


  $[f(x)]^{2} =$$x$
• What is the final volume when x = y^{2}</latex> is revolved around the y-axis from 0 to 2?

  $V =$$\frac{32\pi}{5}$
• Match the concept with its definition:

  Disc Method ↔️ Calculates volume by slicing solid into discs

  Revolution around x-axis ↔️ Region revolves horizontally

  Revolution around y-axis ↔️ Region revolves vertically
• When $y =$$x^{2}$ is revolved around the x-axis from 0 to 1, the final volume is $\frac{\pi}{5}$
• The formula for revolving around the x-axis is $V =$$\pi \int_{a}^{b} [f(x)]^{2} dx$
• What formula is used to find the volume when a region under the curve $y =$$f(x)$ is revolved around the x-axis?


  $V =$$\pi \int_{a}^{b} [f(x)]^{2} dx$
• Revolving a region under a curve around the x-axis creates a solid
• Steps to calculate the volume of a solid formed by revolving a region around the x-axis

  1️⃣ Square the function

  2️⃣ Set up the integral

  3️⃣ Evaluate the integral

  4️⃣ State the final volume
• When a region bounded by $x =$$g(y)$ is revolved around the y-axis, the volume is calculated using $V =$$\pi \int_{c}^{d} [g(y)]^{2} dy$.
• What formula is used to find the volume when a region bounded by $x =$$g(y)$ is revolved around the y-axis?


  V = \pi \int_{c}^{d} [g(y)]^{2} dy</latex>
• The first step in calculating volume using the Disc Method is to square the function
• What is $(x^{2})^{2}$ equal to?


  $x^{4}$
• Forgetting to square the function is a common mistake when using the Disc Method.