Washer Method Calculator

What is the Washer Method?

The Washer Method is a technique in calculus used to find the volume of a solid of revolution when the region being revolved does NOT touch the axis of rotation. This creates a solid with a hole in the middle, and the cross-sections perpendicular to the axis are washers (annular rings).

The method extends the Disk Method by accounting for the hole. You calculate the volume as the integral of the area of washers: π times (outer radius² - inner radius²).

Washer Method Formula

When to Use Washer Method vs. Disk Method

1. Washer Method (Use when there's a HOLE)

Characteristics:

• Region doesn't touch axis of rotation
• Cross-sections are rings (washers)
• Need TWO radius functions
• Example: Area between y=x² and y=x revolved around x-axis

2. Disk Method (Use when NO HOLE)

Characteristics:

• Region touches axis of rotation
• Cross-sections are solid disks
• Need ONLY ONE radius function
• Example: Area under y=√x from 0 to 4 revolved around x-axis

3. Shell Method (Alternative)

Characteristics:

• Revolve around axis PARALLEL to slice
• Uses cylindrical shells
• Often easier for y-axis revolution
• Formula: V = 2π ∫ radius × height dx

Step-by-Step Washer Method Process

Step 1: Identify the Region

• Sketch the curves: Draw both functions and identify the area between them
• Determine limits of integration: Find where the curves intersect (solve f(x) = g(x))
• Identify axis of revolution: Determine which axis (x, y, or other line) you're revolving around

Step 2: Determine Radii

• Outer radius (R): Distance from axis of revolution to the FURTHER curve
• Inner radius (r): Distance from axis of revolution to the CLOSER curve
• Important: R(x) ≥ r(x) for all x in [a,b] (or R(y) ≥ r(y) for all y in [c,d])

Step 3: Set Up the Integral

• Square both radii: Calculate R(x)² and r(x)² (or R(y)² and r(y)²)
• Subtract: Form the integrand [R(x)² - r(x)²] (or [R(y)² - r(y)²])
• Multiply by π: The integrand becomes π[R(x)² - r(x)²]
• Add integration limits: Use the intersection points as limits

Step 4: Evaluate the Integral

• Simplify the integrand: Expand and combine like terms if possible
• Integrate: Find the antiderivative of the integrand
• Apply Fundamental Theorem: Evaluate F(b) - F(a) (or F(d) - F(c))
• Include π: Multiply the result by π to get the volume

Common Examples and Applications

Functions	Limits	Axis	Volume Formula	Application
R(x) = 4, r(x) = 2	[0, 5]	x-axis	V = π∫[0,5] (16-4)dx = 60π	Hollow cylinder (pipe)
R(x) = x+2, r(x) = x	[0, 3]	x-axis	V = π∫[0,3] ((x+2)²-x²)dx	Tapered hollow solid
R(y) = √y, r(y) = y/2	[0, 4]	y-axis	V = π∫[0,4] (y - y²/4)dy	Bowl-shaped vessel
R(x) = sin(x)+2, r(x) = 1	[0, π]	x-axis	V = π∫[0,π] ((sin(x)+2)²-1)dx	Wavy-walled container

Washer Method Properties and Tips

Property	Description	Example	Common Mistake to Avoid
Radius Determination	R(x) is ALWAYS ≥ r(x) for all x in interval	For area between y=4 and y=x² from x=0 to 2: R(x)=4, r(x)=x²	Switching inner and outer radii
Limits of Integration	Must be intersection points or given boundaries	For y=x and y=x²: solve x=x² → x=0,1	Using wrong intersection points
Squaring Radii	Square FIRST, then subtract: (R(x))² - (r(x))²	NOT (R(x) - r(x))² which is incorrect	Subtracting radii before squaring
Axis of Revolution	Determines variable of integration (dx vs dy)	x-axis → dx, y-axis → dy	Using wrong differential (dx/dy)

Detailed Example Walkthrough

Example: Find volume of solid from region between y=x² and y=x revolved around x-axis, x=1 to x=2

1. Identify region: Between y=x² (parabola) and y=x (line), from x=1 to x=2
2. Determine radii: When revolving around x-axis:
   • Outer radius R(x) = x (the line is ABOVE the parabola in this interval)
   • Inner radius r(x) = x² (the parabola is BELOW the line)
3. Set up integral: V = π ∫[1,2] [(x)² - (x²)²] dx = π ∫[1,2] [x² - x⁴] dx
4. Evaluate integral:
   • Antiderivative: F(x) = π [x³/3 - x⁵/5]
   • F(2) = π [8/3 - 32/5] = π [40/15 - 96/15] = π(-56/15)
   • F(1) = π [1/3 - 1/5] = π [5/15 - 3/15] = π(2/15)
   • V = F(2) - F(1) = π(-56/15 - 2/15) = π(-58/15) = 58π/15 (take absolute value)
5. Final volume: V = (58π)/15 ≈ 12.148 cubic units

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Frequently Asked Questions (FAQs)

Q: What's the difference between washer method and disk method?

A: The disk method is used when the region touches the axis of revolution (creating solid disks). The washer method is used when there's a gap between the region and axis (creating washers/rings with holes). Washer method formula has R² - r² while disk method has just R².

Q: How do I know which function is outer radius and which is inner radius?

A: For revolution around x-axis: outer radius is the function with LARGER y-values in the interval. For revolution around y-axis: outer radius is the function with LARGER x-values. Always sketch the region to visualize which curve is further from the axis.

Q: Can I use washer method for revolution around y-axis?

A: Yes! The formula becomes V = π ∫[c,d] [R(y)² - r(y)²] dy where R(y) and r(y) are functions of y, and you integrate with respect to y. The calculator above supports both x-axis and y-axis revolution.

Q: What if my functions cross in the interval?

A: If functions cross, the outer/inner roles switch at the intersection point. You must split the integral at that point, using R(x) as the top function and r(x) as the bottom function on each subinterval separately.

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