Welcome back to class! In our session on Section 6.2, we moved from calculating 2D areas to finding the Volumes of 3D solids. This is one of the most visual and exciting applications of integration in Calculus II. We focused specifically on solids of revolution, where we rotate a region around an axis to create a shape.

1. The General Definition

At its core, volume is calculated by integrating the area of cross-sections perpendicular to an axis. If $S$ is a solid lying between $x=a$ and $x=b$, and $A(x)$ is the cross-sectional area, the volume is defined as:

$$ V = \int_a^b A(x) \, dx $$

This is the foundation for the two specific methods we practiced in class.

2. The Disk Method

When the region we are rotating is fully flush against the axis of rotation (creating a solid with no holes), we use the Disk Method. Here, our cross-sections are circles. Since the area of a circle is $A = \pi r^2$, our integral becomes:

$$ V = \pi \int_a^b [f(x)]^2 \, dx $$

Class Example: We derived the volume of a sphere! By rotating the semicircle $y = \sqrt{r^2-x^2}$ around the x-axis, we integrated from $-r$ to $r$. The algebra beautifully simplifies to the famous formula $V = \frac{4}{3}\pi r^3$.

3. The Washer Method

Things get interesting when there is a gap between our region and the axis of rotation. When rotated, this gap creates a hole, turning our cross-sections into "washers" (rings) rather than solid disks. To find the area of a washer, we subtract the area of the inner circle from the outer circle:

$$ V = \pi \int_a^b \left( [R_{outer}(x)]^2 - [r_{inner}(x)]^2 \right) \, dx $$

Key Strategy: Always draw a "representative slice" on your graph. Connect it to the axis of rotation to identify your Outer Radius ($R$) and Inner Radius ($r$).

Important Takeaways from the Notes

  • Rotation Axis Matters: We looked at examples rotating around the y-axis (like the region bounded by $y=x^3$ and $y=8$). In these cases, remember to solve for $x$ in terms of $y$ and integrate with respect to $y$ ($dy$).
  • Shifted Axes: We solved a complex problem rotating the region between $y=\sin x$ and $y=\cos x$ around the line $y=-1$. This changed our radii to $(y_{curve} - (-1))$, or $(y_{curve} + 1)$.
  • Trig Identities: Don't forget your identities! In the sine/cosine example, we used the double angle identity to simplify integration: $$ \cos^2 x - \sin^2 x = \cos(2x) $$

Mastering volume requires visualizing the 3D shape and carefully setting up your radii. Keep practicing those integrals, and I'll see you in the next class!