How to Find Volume by Spinning: Washer Method
TL;DR: Spin a region that does not touch the axis and you get a solid with a hole down the middle, like a stack of washers. The washer method handles exactly this case. Each slice is an outer disk minus an inner disk, so you integrate pi times outer radius squared minus inner radius squared from a to b. Use it whenever the region floats away from the axis of rotation. Two radii, one subtraction, you have got it.
Key takeaways:
- Washer method: \(V=\pi\int_a^b ([R(x)]^2-[r(x)]^2)\,dx\).
- Each washer is a disk with a hole - outer radius \(R\), inner radius \(r\).
- Use washers when there's a gap between the region and the axis of rotation.
- Outer radius and inner radius are both measured FROM the axis of rotation.
- If \(r(x) = 0\), the washer collapses to a regular disk.
The Washer Method extends the Disk Method to calculate volumes of solids with hollow centers by revolving a region around an axis. This technique involves subtracting the volume of the inner solid from the outer solid, integrating the difference between the outer and inner radii squared, multiplied by π, across the interval. Ideal for objects like donuts or rings, it accurately measures volumes of complex, annular shapes.
This method adeptly handles scenarios where solids encase voids, introducing a nuanced layer to volume calculations. It’s invaluable for engineering applications requiring precise volume estimations of cylindrical tanks, pipes, and certain architectural structures with cylindrical cavities.
Mathematical Explanation:
Given two functions, \(y = f(x)\) (outer curve) and \(y = g(x)\) (inner curve), revolving around the x-axis, the volume \(V\) is calculated as:
\( V = \pi \int_{a}^{b} \left( [f(x)]^2 – [g(x)]^2 \right) dx \)
where \([a, b]\) are the bounds of integration. This formula accounts for the space between the curves by considering the area of the outer disk minus the inner disk, effectively capturing the volume of the “washer” shaped slices.
Let’s calculate the volume of the solid formed by revolving the region between the curves \( y = x^2 \) and \( y = x + 2 \) around the x-axis from \( x = 0 \) to \( x = 2 \).
Step-by-Step Solution:
- Identify the Functions and Interval:
- Outer function (larger values on the interval): \( y = x + 2 \)
- Inner function (smaller values on the interval): \( y = x^2 \)
- Interval: \( x = 0 \) to \( x = 2 \)
- Set Up the Integral for the Washer Method:
- The volume \( V \) is calculated by integrating the difference between the squares of the outer and inner functions, multiplied by \( π \), over the given interval:
\( V = \pi \int_{0}^{2} [(x + 2)^2 – (x^2)^2] dx \)
- Evaluate the Integral:
- Simplify and integrate:
\( V = \pi \int_{0}^{2} [x^2 + 4x + 4 – x^4] dx \)
The volume of the solid formed by revolving the region between the curves \( y = x^2 \) and \( y = x + 2 \) around the x-axis from \( x = 0 \) to \( x = 2 \) is approximately \(38.54\) cubic units.
Recommended EffortlessMath Books
For the calculus background behind volumes of revolution, the Pre-Calculus for Beginners pins down the function and integration prerequisites, and the Calculus for Beginners walks through every integration technique – including disk, washer, and shell methods – with full worked examples.
Frequently Asked Questions
What is the washer method in calculus?
The washer method finds the volume of a solid made by spinning a region that does not touch the axis of rotation. Each slice is a disk with a hole, so its area is π(R² − r²), where R is the outer radius and r is the inner radius. Adding up all the slices gives the formula V = π∫ₐᵇ ([R(x)]² − [r(x)]²) dx.
How do you do the washer method?
Sketch the region and the axis, then find the bounds by setting the two curves equal. Name the outer radius R (axis to the farther curve) and the inner radius r (axis to the closer curve), and integrate π(R² − r²). For y = x and y = x² about the x-axis, the bounds are 0 and 1, so V = π∫₀¹ (x² − x⁴) dx = π(1/3 − 1/5) = 2π/15.
How do you do the washer method around a line?
Measure both radii from the line, not from the x-axis. Rotating the region between y = x and y = x² (0 ≤ x ≤ 1) about y = −1, the outer radius is x + 1 and the inner radius is x² + 1, so V = π∫₀¹ ((x + 1)² − (x² + 1)²) dx. For a line above the region, such as y = 2, the radii become 2 − x² (outer) and 2 − x (inner).
Is the disk and washer method the same?
They are closely related but not identical. The disk method uses one radius because the region touches the axis, giving V = π∫R² dx. The washer method subtracts a hole, giving V = π∫(R² − r²) dx. If the inner radius is 0, the washer formula turns into the disk formula, so the disk method is really a special case of washers.
Is the washer method in AP Calc AB?
Yes. Finding volumes of solids of revolution with disks and washers is part of AP Calculus AB, and it appears in BC as well, within the applications of integration unit. In college it usually comes up in Calculus 2. Expect problems that rotate a bounded region about the x-axis, the y-axis or another horizontal or vertical line.
What is the difference between the washer method and the shell method?
Washers slice the solid perpendicular to the axis of rotation, while shells slice parallel to it. For rotation about the x-axis, washers integrate in x and shells integrate in y. Both give the same volume for the same region, so pick the setup with simpler radii and bounds. Washers are usually easier when the curves are written as y = f(x) and you spin about a horizontal line.
Related EffortlessMath Lessons
If a topic on this page feels rusty, these short lessons go deeper:
Related to This Article
More math articles
- What Socialization Is and Is Not
- 5th Grade MAP Math Practice Test Questions
- How to Do Division Using Repeated Subtraction
- Delaware Smarter Balanced Grade 7 Math Worksheets: 95 Free Single-Skill PDFs
- Panics of 1819 and 1837: Banking, Credit and Specie Policy
- The Ultimate SHSAT Math Formula Cheat Sheet
- Identify 3–Dimensional Figures
- Rules of Exponents
- What Skills Do I Need for the SAT Math Test?
- How to Use Measures of Center and Spread to Compare Populations
What people say about "Washer Method: Volume of a Solid of Revolution (Formula)"?
No one replied yet.