Washer Method Calculator

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):

Upper limit:

Lower limit:

Equation Preview

Washer Method Calculator:

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.

What Is Washer Method?

“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.”

Washer Method Formula:

The exact volume formula arises from taking a limit as the number of slices becomes infinite.

Formula for washer method is:

V = π ∫_a^b[f(x)² - g(x)²] dx

The Washer Method (Step-by-Step):

Write down the outer function (f(x)) (further from the axis) and the inner function (g(x))
Now find the limits of integration ((a) and (b))
Set up the integral: (V = π ∫_a^b[f(x)² – g(x)²] dx)(for revolution around a horizontal axis), Adjust if revolving around a vertical axis
Evaluate the integral
Calculate the final volume (V)

Example:

Use the washer method to find the exact volume of the solid formed by rotating the region bounded with the functions:

f(x) = x + 2 and g(x) = 2
from x = 2 to x = 3
about the x-axis

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

How To Use the Washer Method Calculator?

Follow these steps to accurately find volume using washer method calculator:

Enter Functions: Enter the value for the functions f(x) and g(x)
Enter Limits: Substitute the upper and lower limit
Click on the “Calculate” Button: Once you have provided all the required information in the designated fields, click on the calculate button to start the volume calculation
View Results: Our calculator will show the indefinite and definite integral with a detailed breakdown of each step

FAQ’s:

When to Use the Washer Method?

Washer Method: It is used when there are two or more curves and you look at the area between them
Disk Method: The disk method is used when the area under the curve extends to the axis of rotation

What is the Difference Between the Disk and Washer Method?

These both methods are used to calculate the volume of a solid of revolution by integrating the areas of cross-sectional slices.

Disk Method:

It is used when the solid has no hole in the center
When the region is rotating around an axis and is cut into perpendicular slice
When there is no gap between the region and the axis of revolution
All slices are in the form of a disk

Washer Method:

It is used when the rotating solid has a hole in the middle
When there is a gap between the region and axis of rotation
All the slices are in the form of disk having hole in the center

How Do We Tell If It's a Disk or Washer?

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.

What Are Common Mistakes to Avoid When Using the Washer Method?

Wrong identification of outer or inner radii
Use of incorrect limits for integration
Algebraic errors while performing the integration

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.