📚 Topic Summary
The Disk Method is a technique in calculus used to find the volume of a solid of revolution. We imagine slicing the solid into thin disks perpendicular to the axis of rotation. The volume of each disk is approximately the area of the circle times the thickness ($πr^2h$). By integrating these volumes over the interval of rotation, we obtain the total volume of the solid. When rotating around the x-axis, the radius, $r$, is typically a function of $x$, and the thickness, $h$, is $dx$. Therefore, the general formula for the volume using the disk method around the x-axis is: $V = \int_{a}^{b} π[f(x)]^2 dx$, where $f(x)$ defines the curve being rotated, and $a$ and $b$ are the limits of integration along the x-axis.
🧠 Part A: Vocabulary
Match the terms with their definitions:
| Term |
Definition |
| 1. Solid of Revolution |
A. The line around which a 2D shape is rotated. |
| 2. Disk Method |
B. A calculus technique to find the volume of a 3D shape. |
| 3. Radius |
C. The distance from the axis of revolution to the curve. |
| 4. Axis of Revolution |
D. The 3D shape formed by rotating a 2D shape around an axis. |
| 5. Volume |
E. The amount of space occupied by a 3D object. |
✍️ Part B: Fill in the Blanks
The Disk Method is used to find the ________ of a solid formed by revolving a region around an axis. We integrate the area of ________ along the axis of revolution. If we rotate around the x-axis, the integral is with respect to ________, and the radius is a function of ________.
🤔 Part C: Critical Thinking
Explain, in your own words, why the disk method involves integration and how it relates to summing up infinitely thin slices to find the total volume. Provide a simple real-world analogy to support your explanation.