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### LEARNING OBJECTIVE CHA-5.C: Calculate Volumes of Solids of Revolution Using Definite Integrals

**ESSENTIAL KNOWLEDGE CHA-5.C.2**: Volumes of solids of revolution around any horizontal or vertical line in the plane may be found by using definite integrals with the disc method.

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### Key Concept:

The **disc method** is a technique for calculating the volume of a solid generated by rotating a region in the plane around an axis (either horizontal or vertical). This method is applicable even when the axis of revolution is not necessarily one of the coordinate axes (i.e., not limited to the x-axis or y-axis).

### Volumes Around Horizontal or Vertical Lines:

- **Horizontal or Vertical Line as Axis of Revolution**:
A solid of revolution is formed by rotating a region about any horizontal or vertical line, such as \( y = c \) (horizontal) or \( x = c \) (vertical), where \( c \) is a constant.

### The Disc Method:

1. **For Revolving Around a Vertical Line** (e.g., \( x = c \)):
The radius of the disc at a given point is the horizontal distance between the curve and the vertical axis of revolution. If the region is bounded by \( y = f(x) \), the radius of each disc will be the distance from the curve to the axis \( x = c \), or \( |x - c| \). The volume is then found using the integral:
\[
V = \pi \int_{a}^{b} \left( f(x) - c \right)^2 \, dx
\]
where \( a \) and \( b \) are the bounds of the region in the \( x \)-direction.

2. **For Revolving Around a Horizontal Line** (e.g., \( y = c \)):
The radius of the disc at a given point is the vertical distance between the curve and the horizontal axis of revolution. If the region is bounded by \( x = g(y) \), the radius of each disc is \( |y - c| \). The volume is then given by the integral:
\[
V = \pi \int_{c_1}^{c_2} \left( g(y) - c \right)^2 \, dy
\]
where \( c_1 \) and \( c_2 \) are the bounds of the region in the \( y \)-direction.

### Practical Application:

- **Flexibility of the Disc Method**:
The disc method can be applied to any solid of revolution formed by rotating a region around a vertical or horizontal line, making it a versatile tool in volume calculation. Whether the axis of revolution is the x-axis, y-axis, or any line parallel to them, the disc method allows you to compute the volume of the resulting solid.

- **Example Problem**:
For a region bounded by \( y = f(x) \) and rotated around a vertical line \( x = c \), the volume is given by:
\[
V = \pi \int_{a}^{b} (f(x) - c)^2 \, dx
\]
If \( y = x^2 \) and we are rotating around \( x = 2 \), then the radius at a given point would be \( 2 - x \), and the volume integral would be:
\[
V = \pi \int_{0}^{2} (2 - x)^2 \, dx
\]

### Conclusion:

CHA-5.C.2 emphasizes that the disc method is a powerful technique to calculate volumes of solids of revolution when the axis of revolution is any horizontal or vertical line in the plane. By setting up the appropriate definite integral for the volume based on the radius of the discs, the method can be applied to a wide variety of scenarios, making it an essential tool for solving problems in calculus related to solids of revolution.

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Nick Perich
Norristown Area High School
Norristown Area School District
Norristown, Pa

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