Calculus: Washer Method Quiz

Introduction to the Washer Method

• The washer method is a calculus technique for finding the volume of a solid of revolution.
• It's useful for solids with a hole in the middle, formed when a region is rotated around an axis.
• This method builds on the disk method, accounting for the area between the inner and outer radii.

Understanding the Concept

• The method works with a region in the xy-plane enclosed by curves.
• This region rotates around a horizontal or vertical axis.
• The goal is to calculate the volume of the resulting solid.
• The solid is sliced into infinitely thin washers (disks with holes).
• Each washer's outer radius is the distance from the axis to the outermost curve.
• Each washer's inner radius is the distance from the axis to the innermost curve.
• The washer area is π(outer radius²) - π(inner radius²).

Setting up the Integral

• The volume is found by integrating the washer areas over the integration interval.
• The integral depends on the revolution axis and the region boundaries in the xy-plane.

Key Differences to the Disk Method

• The disk method applies when the region touches the rotation axis at all points.
• The washer method applies when the region is separated from the axis, creating a gap (hole).
• The difference lies in the calculation; washers account for the inner radius.

Formulas and Variables

• Volume: ∫_a^b π(R(x)² - r(x)² ) dx (horizontal axis). Use 'dy' for a vertical axis.

• R(x): Outer radius, function of x (distance to outermost curve from the axis).

• r(x): Inner radius, function of x (distance to innermost curve from the axis).

Choosing the Variable of Integration

• Use 'dx' with a horizontal axis and functions in terms of x.
• Use 'dy' with a vertical axis and functions in terms of y.

Steps for Application

• 1. Sketch the region: Visualize the revolving region.
• 2. Identify curves: Determine the functions defining the region's boundaries.
• 3. Determine the rotation axis: Establish the axis of rotation.
• 4. Find radii: Express outer and inner radii as functions of x or y (depending on the axis).
• 5. Set up the integral: Substitute radius functions into the volume formula.
• 6. Evaluate the integral: Integrate over the appropriate interval.
• 7. Calculate volume: This gives the volume of the solid of revolution.

Example Scenarios

• Revolving y = x², y = 0, x = 0, and x = 2 around the x-axis.
• Rotating y = x, y = √x, x = 1, and x = 4 around the line y = 5.
• Rotating y = x² and y = 2x around the y-axis.

Important Considerations

• Carefully identify outer and inner radii in the integral.
• Precisely determine the limits of integration (a and b) – based on intersection points.
• Thoroughly check every step to avoid calculation errors.
• Employ calculator technology when necessary for complex calculations.