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### **LEARNING OBJECTIVE: CHA-5.C**
**Calculate volumes of solids of revolution using definite integrals.**
### **ESSENTIAL KNOWLEDGE: CHA-5.C.4**
Volumes of solids of revolution around any horizontal or vertical line whose cross-sections are ring-shaped may be found using definite integrals with the **Washer Method**.
### **Key Concept:**
The **Washer Method** is used to calculate the volume of solids of revolution when the solid has **ring-shaped cross sections**. This method applies when a region is revolved around a horizontal or vertical axis, and the cross sections perpendicular to the axis of revolution are washers (disks with a hole in the middle).
### **How the Washer Method Works:**
1. **Identify the region** being revolved and the **axis of revolution**.
2. For each cross section, calculate the **outer radius** (distance from the axis of revolution to the outer curve) and the **inner radius** (distance from the axis of revolution to the inner curve).
3. The area of the washer at each point is the difference between the area of the outer disk (with the outer radius) and the inner disk (with the inner radius).
4. The volume of the solid is then found by integrating the area of these washers across the interval that bounds the region.
### **Washer Method Formula:**
For revolution around a **vertical line** \( x = k \), the volume is given by:
\[
V = \pi \int_{a}^{b} \left[ (R(x))^2 - (r(x))^2 \right] \, dx
\]
where:
- \( R(x) \): the outer radius (distance from the axis \( x = k \) to the outer curve),
- \( r(x) \): the inner radius (distance from the axis \( x = k \) to the inner curve),
- \( a \) and \( b \): the bounds of integration for the region in the \( x \)-direction.
For revolution around a **horizontal line** \( y = k \), the formula becomes:
\[
V = \pi \int_{c}^{d} \left[ (R(y))^2 - (r(y))^2 \right] \, dy
\]
where:
- \( R(y) \): the outer radius (distance from the axis \( y = k \) to the outer curve),
- \( r(y) \): the inner radius (distance from the axis \( y = k \) to the inner curve),
- \( c \) and \( d \): the bounds of integration for the region in the \( y \)-direction.
### **When to Use the Washer Method:**
- Use the washer method when the solid is created by revolving a region around a line, and the resulting cross sections are ring-shaped.
- The method is applicable to both horizontal and vertical lines of revolution.
- It is particularly useful when there is a gap or hole between two curves that define the boundaries of the region being revolved.
### **Example:**
Consider a region bounded by the curves \( y = x^2 \) and \( y = x + 1 \), and suppose this region is revolved around the vertical line \( x = 2 \). To find the volume, you would calculate the outer and inner radii for each washer, then integrate the area of the washers to obtain the total volume.
### **Conclusion:**
CHA-5.C.4 focuses on applying the **Washer Method** to calculate the volume of solids formed by revolving regions around horizontal or vertical axes. By evaluating definite integrals of the differences between the squared outer and inner radii, this method allows for the calculation of volumes of solids with ring-shaped cross sections.
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Nick Perich
Norristown Area High School
Norristown Area School District
Norristown, Pa
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