8.16 Area of a Surface of Revolution

• What is a surface of revolution created by?

  Rotating a curve around an axis
• What does r(x) represent in the surface area formula?

  Distance from axis to curve
• The definition of r(x) changes based on the axis of rotation.


  True
• What is the key factor in defining r(x) when calculating surface area?

  Axis of rotation
• Steps to calculate the surface area of a surface of revolution:

  1️⃣ Identify the axis of rotation

  2️⃣ Define r(x) based on the axis

  3️⃣ Set up the integral using the surface area formula

  4️⃣ Evaluate the integral
• What does $r(x)$ represent in the surface area formula?


  Distance from the axis
• In the surface area formula, r(x) is the distance from the axis of rotation to the curve
• Steps to evaluate the integral for rotation around the x-axis

  1️⃣ Substitute the function for $r(x)$

  2️⃣ Compute the derivative $\frac{dr(x)}{dx}$

  3️⃣ Plug values into the integral

  4️⃣ Evaluate the integral
• A surface of revolution is formed by rotating a curve
• Symmetry can simplify the calculation of surface area


  True
• When rotating around the x-axis, $r(x) =$$y$


  True
• The definition of $r(x)$ changes based on the axis of rotation
• When rotating around the y-axis, $r(x) =$$x$


  True
• Steps to evaluate the surface area when rotating around the y-axis:

  1️⃣ Substitute $r(y)$ into the formula

  2️⃣ Compute the derivative $\frac{dr(y)}{dy}$

  3️⃣ Plug values into the integral

  4️⃣ Evaluate the integral
• The surface area generated by rotating $x =$$\sqrt{y}$ from $y =$$0$ to $y =$$4$ around the y-axis is $\frac{\pi}{6}(17\sqrt{17} - 1)$


  True
• The surface area of a paraboloid bowl y = x^2</latex> from $x =$$0$ to $x =$$2$ around the x-axis is $S =$$\int_{0}^{2} 2\pi x^{2} \sqrt{1 + (2x)^{2}} dx$


  True
• The symmetry of the generated surface simplifies surface area calculations.


  True
• The variables a and b in the surface area formula represent the limits of integration
• Match the axis of rotation with the correct definition of r(x):

  x-axis ↔️ r(x) = y

  y-axis ↔️ r(x) = x
• Understanding the initial curve and axis placement helps visualize the final surface
• When rotating around the x-axis, r(x) is equal to y.


  True
• When rotating around the x-axis, $r(x) =$$y$


  True
• What is $r(x)$ when rotating around the y-axis?


  x
• What is the formula for surface area when rotating around the y-axis?

  $S =$$\int_{a}^{b} 2\pi r(y) \sqrt{1 + \left(\frac{dr(y)}{dy}\right)^{2}} dy$
• Match the axis of rotation with the curve used to generate the surface:

  x-axis ↔️ y = f(x)

  y-axis ↔️ x = g(y)
• a and b are the limits of integration
• Match the axis of rotation with the correct $r(x)$:


  x-axis ↔️ r(x) = y

  y-axis ↔️ r(x) = x
• Steps to evaluate the surface area when rotating around the x-axis:

  1️⃣ Substitute $r(x)$ into the surface area formula

  2️⃣ Compute the derivative $\frac{dr(x)}{dx}$

  3️⃣ Plug values into the integral

  4️⃣ Evaluate the integral
• The formula for surface area when rotating around the y-axis is S = \int_{c}^{d} 2\pi x \sqrt{1 + \left(\frac{dx}{dy}\right)^{2}} dy
• Example: Find the surface area generated by rotating $x =$$\sqrt{y}$ from $y =$$0$ to y = 4</latex> around the y-axis
• Match the application with the correct axis of rotation and surface area calculation:

  Water Tower ↔️ y-axis, $S =$$\int_{0}^{4} 2\pi x \sqrt{1 + \left(\frac{dx}{dy}\right)^{2}} dy$

  Torus ↔️ x-axis, $S =$ \int_{0}^{2\pi} 2\pi (2sin(t)) \sqrt{\left(\frac{dx}{dt}\right)^{2} + \left(\frac{dy}{dt}\right)^{2}} dt