Enter your functions f(x) and g(x), along with the upper and lower limits, to calculate the volume of a solid revolving around an axis. Get a detailed step-by-step solution.
Use this washer method calculator to determine the volume of solids formed by revolution around an axis. By using the integration technique of the washer method, our tool provides accurate, step-by-step calculations.
“The washer method is a technique in calculus, used to determine the volume of a revolving solid bounded by two functions f(x) (outer boundary) and g(x) inner boundary around an axis. It is achieved by performing integration.”
The exact volume formula arises from taking a limit as the number of slices becomes infinite.
Formula for washer method is:
Use the washer method to find the exact volume of the solid formed by rotating the region bounded with the functions:
Solution:
Step #1: Function(s):
f(x) = x + 2
g(x) = 2
Step #2: Interval of Integration:
a = 2, b = 3
Step #3: Outer Radius R(x):
R(x) = f(x) = x + 2
Step #4: Inner Radius r(x):
r(x) = g(x) = 2
Step #5: Volume Formula:
V = π ∫23 ([f(x)]² - [g(x)]²) dx
Step #6: Substitute Functions:
V = π ∫23((x + 2)² - (2)²) dx
Step #7: Simplify Integrand:
V = π ∫23(x² + 4x) dx
Step #8: Integrate:
V = (49π) / 3
Step #9: Final Approximation:
Volume = 51.3126800086333
Follow these steps to accurately find volume using washer method calculator:
These both methods are used to calculate the volume of a solid of revolution by integrating the areas of cross-sectional slices.
Disk Method:
Washer Method:
To tell whether its disk or washer method, check the region which is revolved if it touches the axis of rotation then its disk or if it has a gap then its washer.
References:
Wikipedia: Disc integration, Washer method, the axis of revolution.
Magoosh: Solids of Revolution, The Disk and Washer Methods: Formulas, Different Axes.
Dummies: Volume of a Shape Using the Washer Method, Outer and Inner Radius.
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