Washer Method Calculator

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.

f(x):

g(x):

Lower limit:

Upper limit:

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Washer Method Calculator

Use this calculator to determine the volume of solids formed by revolution around an axis. By applying the washer method in calculus, the tool provides accurate, step-by-step calculations.

What is the Washer Method?

The washer method is a technique in calculus used to find the volume of a solid of revolution. It applies when a region is bounded by two functions, f(x) (outer boundary) and g(x) (inner boundary), revolved around an axis. Integration is used to sum up infinitesimally thin slices (washers).

Washer Method Formula

V = π ∫_a^b [f(x)² - g(x)²] dx

Where:

f(x) = outer radius (distance from axis to outer curve)
g(x) = inner radius (distance from axis to inner curve)
a, b = limits of integration

Step-by-Step Washer Method

Identify the outer function f(x) and inner function g(x).
Determine the limits of integration a and b.
Set up the integral: V = π ∫[a to b] (f(x)² - g(x)²) dx. Adjust if revolving around a vertical axis.
Evaluate the integral.
Compute the final volume V.

Example:

Find the volume of the solid formed by rotating the region bounded by:

f(x) = x + 2, g(x) = 2
x ∈ [2, 3]
About the x-axis

Solution:

Outer radius: R(x) = f(x) = x + 2
Inner radius: r(x) = g(x) = 2
Limits of integration: a = 2, b = 3
Volume integral: V = π ∫₂³ [(x + 2)² - (2)²] dx
Simplify integrand: V = π ∫₂³ (x² + 4x) dx
Integrate: V = π [x³/3 + 2x²]₂³ = (49π)/3
Approximation: V ≈ 51.31 units³

How to Use the Washer Method Calculator

Enter Functions: Input f(x) and g(x).
Enter Limits: Set the lower and upper bounds a and b.
Calculate: Click "Calculate" to evaluate the integral.
View Results: The calculator provides the indefinite and definite integral along with step-by-step details.

FAQ

When to Use the Washer Method?

Use the washer method when there is a gap between the solid and the axis of rotation (a hole in the middle).
Use the disk method when the solid extends all the way to the axis (no hole).