Welcome back to class! In our previous lesson, we learned how to measure the length of a curve using Arc Length. Tonight, in Section 8-2, we take that concept one dimension higher. By taking a curve and rotating it around an axis, we create a 3D shell, and our goal is to calculate the Area of a Surface of Revolution.

The Concept: From Frustums to Integrals

To understand the formula, visualize the "bands" that make up the surface. Just as we used rectangles to approximate area under a curve, we use the frustum of a cone (a cone with the tip cut off) to approximate sections of our surface. When we unroll a cylinder, we get a rectangle with area $A = 2\pi r h$. Similarly, for a surface of revolution, we sum up infinite tiny bands.

The general formula depends on the radius of rotation ($r$) and the arc length segment ($ds$):

$$S = \int 2\pi r \, ds$$

Key Formulas for Rotation

As detailed in the lecture slides and class notes, the setup changes slightly depending on which axis you are rotating around. Remember that the radius ($r$) is always the distance from the function to the axis of rotation.

  • Rotation about the x-axis: The radius is the height of the function, $y$. $$S = \int_{a}^{b} 2\pi y \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx$$
  • Rotation about the y-axis: The radius is the horizontal distance, $x$. $$S = \int_{a}^{b} 2\pi x \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx$$

Highlights from the Class Notes

The attached handwritten notes cover several crucial examples that you should review closely:

  1. The Sphere (Example 1): We prove that the surface area of a sphere is $4\pi r^2$ by rotating the semicircle $y = \sqrt{4-x^2}$. This is a great way to verify that our calculus matches standard geometry formulas.
  2. The Parabola (Example 2): Rotating $y=x^2$ about the y-axis requires careful u-substitution ($u = 1+4x^2$). Don't forget to adjust your bounds when you change variables!
  3. The Exponential "Beast" (Example 3): Rotating $y=e^x$ leads to a tricky integral involving $\sqrt{1+e^{2x}}$. This example requires a trigonometric substitution ($u = \tan \theta$) and results in the integral of $\sec^3 \theta$.

Mathematical Tip: The $\sec^3 \theta$ Integral

As noted in the final pages of the handwritten notes, $\int \sec^3 \theta \, d\theta$ is a classic, challenging integral that appears frequently in this chapter. It requires Integration by Parts and a recursive loop to solve. Make sure to review the derivation in the notes—it is a powerful technique to have in your toolkit!

Check out the Zoom recording and the PDFs below for the full step-by-step solutions. Happy integrating!