Volumes of Revolution: Disk and Washer Methods

General Concept:
- Volumes of revolution are calculated by integrating the area of cross-sections over a specific interval.
- When an area is rotated around an axis, the resulting cross-section is a circle.
- The fundamental formula used for the area of the base (cross-section) is the area of a circle: $A = \pi r^2$ .
- The integral accumulates these cross-sectional areas multiplied by a small thickness ( $dx$ or $dy$ ) to find the total volume.
Simplifying Visualization:
- It is not necessary to draw a complex 3D representation of the volume.
- The primary goal is to determine the radius ( $r$ ) of the rotation by looking at a 2D sketch of the area being rotated.

The Disk Method:
- Used when the area being rotated is flush against the axis of rotation, resulting in a solid figure with no holes.
- The cross-section is a single solid disk.
- Area: $A = \pi r^2$ .
The Washer Method:
- Used when there is a gap between the area being rotated and the axis of rotation, or when the region is bounded by two different functions.
- This results in a hollow interior, and the cross-section looks like a "washer" (a large circle with a smaller circle removed from the middle).
- To find the area of a washer, you subtract the area of the inner circle from the area of the outer circle:
  - $Area_{washer} = Area_{large} - Area_{small}$
  - $Area_{washer} = \pi (R_{outer})^2 - \pi (r_{inner})^2$
- Wait until the end of the setup to factor out $\pi$ as a constant to simplify calculations.

The Labeling Technique:
1. Identify the Axis of Rotation: Locate the line the figure is spinning around (e.g., the x-axis, the y-axis, or a line like $y = -2$ ).
2. Draw a Path: Start at the axis of rotation and draw a line segment perpendicularly through the area being rotated.
3. Identify Key Points: Identify and label the coordinates of the points where this line segment intersects the axis and the boundaries of the region.
4. Determine Radii:
  - The radius is the distance from the axis of rotation to the function.
  - For horizontal axes: $Radius = |y_{upper} - y_{lower}|$
  - For vertical axes: $Radius = |x_{right} - x_{left}|$

Scenario 1: Rotating about the x-axis ( $y = 0$ ):
- Region: Bounded by $y = x$ and $y = x^2$ from $x = 0$ to $x = 1$ .
- Point on axis: $(x, 0)$ .
- Point on inner boundary: $(x, x^2)$ .
- Point on outer boundary: $(x, x)$ .
- Radii Calculation:
  - Large Radius ( $R$ ): $x - 0 = x$
  - Small Radius ( $r$ ): $x^2 - 0 = x^2$
- Integral Setup:
  - $V = \int_0^1 (\pi (x)^2 - \pi (x^2)^2) \,dx$
  - Simplified: $V = \pi \int_0^1 (x^2 - x^4) \,dx$
Scenario 2: Rotating about a line below the axis ( $y = -2$ ):
- This creates a larger hollow center.
- Point on axis: $(x, -2)$ .
- Point on inner boundary: $(x, x^2)$ .
- Point on outer boundary: $(x, x)$ .
- Radii Calculation:
  - The goal is to find the length (a positive number) by subtracting the bottom value from the top value.
  - Large Radius ( $R$ ): $x - (-2) = x + 2$
  - Small Radius ( $r$ ): $x^2 - (-2) = x^2 + 2$
- Integral Setup:
  - $V = \pi \int_0^1 ((x + 2)^2 - (x^2 + 2)^2) \,dx$

Integration with Respect to y:
- When rotating about the y-axis, functions must be solved for $x$ in terms of $y$ .
- Bounds for the integral are determined by the interval on the y-axis.
Conversion of Functions:
- $y = x \rightarrow x = y$
- $y = x^2 \rightarrow x = \pm\sqrt{y}$ . Since the region is in the first quadrant where $x$ is positive, we use $x = \sqrt{y}$ .
Integral Setup for y-axis Rotation:
- Interval: $y$ goes from $0$ to $1$ .
- Point on axis: $(0, y)$ .
- Outer Point: $(\sqrt{y}, y)$ (because $\sqrt{y} > y$ for values between $0$ and $1$ ).
- Inner Point: $(y, y)$ .
- Radii Calculation:
  - Large Radius ( $R$ ): $\sqrt{y} - 0 = \sqrt{y}$
  - Small Radius ( $r$ ): $y - 0 = y$
- Integral:
  - $V = \pi \int_0^1 ((\sqrt{y})^2 - (y)^2) \,dy$
  - Simplified: $V = \pi \int_0^1 (y - y^2) \,dy$

Avoid Common Pitfalls:
- When squaring binomials like $(x + 2)^2$ , do not simply square each term (i.e., $(x + 2)^2 \neq x^2 + 4$ ). This is a frequent error that results in missing the middle term.
- Always use FOIL or the binomial expansion formula: $(a \pm b)^2 = a^2 \pm 2ab + b^2$ .
Step-by-Step for Integration:
1. Set up the radii using coordinate labeling.
2. Write the full integral including $\pi$ .
3. Simplify the integrand algebraically (FOIL and combine like terms) before attempting to integrate.
4. Apply the Power Rule for integration.

Question: For the radius, couldn't you just use $x - x^2$ ?
Response: No, because the radius is measured relative to the axis of rotation, not just the distance between the two functions. If the axis isn't the x-axis (e.g., if it is $y = -2$ ), the radius includes that extra distance.
Question: Would it be wrong to switch $x$ and $y$ conceptually?
Response: You can reflect coordinates about the line $y = x$ , but you must be extremely careful to do it correctly. It is generally safer to set it up based on the actual given axis of rotation.
Discussion on Signs: When subtracting to find the radius, always ensure the result is positive by subtracting the lower value from the upper value (for horizontal axes) or the leftmost value from the rightmost value (for vertical axes). Between $0$ and $1$ , $y$ is larger than $y^2$ , which ensures a positive area/volume.