How to Find Volume by Spinning: Washer Method
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.
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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.
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