Washer Method Calculator

Explore the Washer Method Calculator for precise volume calculations of solids with internal voids. Simplify complex integrations and save time on imperfect cylindrical shapes.

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Introduction to Washer Method Calculator:

Washer method calculator is an amazing source that helps you to find the revolution of a solid that has a hole inside it. Our tool is used to evaluate the volume of revolution of hollow cylindrical shapes using the integration method.

Washer Method Calculator with Steps

The volume of revolution calculator is a helpful tool as it saves the time that you consume in calculating the volume of a given function. It gives you the solution of an imperfect cylindrical shape in some seconds.

What is the Washer Method?

Washer method is a process that is used to find the volume of revolution of solids of a sphere which is hollow inside in a cross-sectional area. The washer method deals with the imperfect sphere shape that has two radii, the inner radii and the outer radii of the sphere.

The washer method used the integration method to solve the given function. It is used to find the volume of that shapes which does not have a perfect cylindrical sphere like a cup. It has a curve so these types of shapes of volume is calculated using the washer method.

Washer Method Formula:

The formula of washer method consists of two radius because of its hollow sphere shape. The formula used by the washer method calculator is as follows,

$$ \pi \int_a^b \biggr(R(x)^2 - r(x)^2 \biggr) dx $$

Whereas,

R(x) = the outer radii
r(x) = the inner radii
V = the volume of a hollow sphere
dx = the integration with respect to x

How to Calculate Volume Using Washer Method?

The solid of revolution calculator calculates the volume of that sphere which is not perfectly cylindrical in three-dimensional space in a bounded region.

To find the volume, the volumes by slicing calculator uses the integration method. The calculation steps of the washer method is:

Step 1:

The washer volume calculator identifies the functions of the solid, the interval over the rotation, and the axis along which it revolves from the given area.

Step 2:

Determine the outer and inner radii as the outer radius R(x) along the axis of rotation to the outside of the sphere and the inner radius r(x) is the distance from the axis of rotation to the inside of the sphere.

Step 3:

The volume of each washer is given by the difference in the volumes of the two disks but its axis may be horizontal or vertical. If the rotation is along x-axis,

$$ V \;=\; \int_a^b \pi [R(x)^2 - r(x)^2 ] dx $$

If the rotation is along y-axis,

$$ V \;=\; \int_a^b \pi [R(y)^2 - r(y)^2] dy $$

Step 4:

Put the value of the inner radii and outer radii and for limits a and b equate both the values of radii to get the upper and lower limit.

Step 5:

Solve the expression after putting the value in the washer method formula to make the integration process easy.

Step 6:

Integrate the function with respect to the given variable of integration.

Step 7:

After integration apply the upper and the lower limits to get an accurate solution of the washer method problem.

Practical Example of Washer Method:

An example of washer method with a solution is given below to understand the calculation process of washer method calculator.

Example: Find the volume of the solid formed by rotating the region bounded by y = x² - 3x + 2 and y = 2x - 1 about the x-axis.

Solution:

Given data is:

For these limits solve both the equation by equating them,

$$ r(x) \;=\; x^2 - 2x + 2,\; R(x) \;=\; 2x - 1\; x \;=\; 1,\; x \;=\; 3 $$

The washer method formula is,

$$ V \;=\; \pi \int_a^b \biggr(R(x)^2 - r(x)^2 \biggr) dx $$

Put the value in the washer method formula,

$$ V \;=\; \pi \int_1^3 \biggr((2x - 1)^2 - (x^2 - 2x + 2)^2 \biggr) dx $$

Simplify the given radii r(x) and R(x) by opening the square root to make the integration process easier,

$$ =\; \pi \int_1^3 (-x^4 + 4x^3 - 4x^2 + 4x - 3) dx $$

Integrate the function with respect to x,

$$ =\; \pi \biggr[ -\frac{1}{5}x^5 + x^4 - \frac{4}{3}x^3 + 2x^2 - 3x \biggr] \biggr|_1^3 $$

Apply the upper and lower limits to get the volume of the given region.

$$ =\; \frac{104}{15} \pi \approx 21.78\; units^3 $$

How to Use the Washer Method Calculator?

The volume of revolution calculator has an easy-to-use interface, so you can easily use it to evaluate the volume of a slightly cylindrical sphere. Before adding the input for the solutions of given washer method problems, you must follow some simple steps. These steps are:

Enter the inner and outer radii of the given region.
Add the variable of integration for the washer method method question.
Enter the upper and the lower limit definite integral in the input box.
Review your input value before hitting the calculate button of solid of revolution calculator to start the calculation process.
Click on the “Calculate” button of the washer calculator to get the desired result of your given washer method problem.
If you want to try our volumes by slicing calculator to check its accuracy then use the load example option.
Click on the “Recalculate” button to get a new page for solving more washer method questions.

Results from Volume of Revolution Calculator:

Washer method calculator gives you the solution to a given washer method problem when you add the input to it. It provides you with solutions that contain as:

Result Option:

When you click on the result option then the washer calculator provides you with a solution for the volume of an imperfect sphere shape questions.

Possible Step:

When you click on the possible steps option it provides you with the solution of the washer method problem where all calculation steps are included.

Benefits of Solid of Revolution Calculator:

The washer volume calculator gives you multiple benefits whenever you use it to calculate the volume of a hollow sphere and get its solution immediately. These benefits are:

Washer method formula calculator is a free-of-cost tool that provides you the solution for a given vector to find the volume of rotation of a sphere that has a hole inside it without paying a single penny.
It is an adaptive tool that allows you to find the solution of various types of imperfect cylindrical shapes of the given region in three dimensions.
You can use this washers method calculator for practice so that you get a stronghold of this concept.
The volume of revolution calculator is a trustworthy tool that provides you with accurate solutions as per your input to calculate the washer method problem.
Washer method calculator is a speedy tool that provides you solutions of washer method problems in a few seconds.

Related References

How would you define washer method?
Explain the Evaluation process of washer method:
Give some Examples of washer method:
Give an overview of washer method?

Frequently Ask Questions

When i should use washer method to calculate volume

The washer method is a numerical method that is used to find the volume of a solid of revolution. It is a useful process for calculating volumes of solid where you rotate a region around an axis. When to Use the Washer Method

Solid of Revolution:

Use the washer method when you have a region bounded by two curves and you’re rotating this region around an axis to form a solid.

Complex Shapes:

It is used for calculating volumes of solids that cannot be easily find by simple geometric shapes like spheres or cylinders.

Integral Calculus:

This method is used in integral calculus where you need to set up and evaluate integrals to find the volume of solids.

Is the washer method calculating volume

Yes, the washer method is process that used for calculating the volume of solids of revolution. It calculates the volume,revolution of solid objects around its axis, especially for cylindrical objects that is inside round like a coffee cup.

What is the difference between the shell and washer method

The shell and washer method is a numerical integration process for finding the volume of different objects in two or three dimension. Here is the difference of shell or washer method:

Shell Method:

Shell method is a process that used to find the volume of a solid of revolution. It’s useful when the region being rotated has a more straightforward from the axis of rotation.

Each shell has a radius, height, and thickness, and its volume can be approximated as a thin cylindrical ring.

Shell method used for :

Solid of Revolution: A three-dimensional object formed by rotating a two-dimensional shape around an axis.
Cylindrical Shell: A thin, hollow cylinder (shell) with a certain radius, height, and thickness. The volume of the solid is the sum up of the volumes of the shells.
Volume Calculation: The method calculates the volume of each cylindrical shell and then integrating to find the total volume of the solid. Shell Method Formula,

$$ V \;=\; 2 \pi \int_a^b x . f(x) dx $$

Washer Method

The washer method is a numerical integration method that used to find the volume of a solid of revolution. It calculates the volume of objects with a central hollow region, such as rings or hollow cylinders. Here's a detailed the washer method,

Formula:
When rotating around the x-axis:

$$ V \;=\; \pi \int_a^b [(R(x))^2 - (r(x))^2] dx $$

The washer method is ideal for computing volumes when the outer and inner curves are used for the functions along axis of rotation in a solid like a cylindrical hole.

How to use the washer method in calculus

The washer method in calculus is a process for finding the volume of a solid of revolution. It is useful when the solid has a hollow shaped, when you need to find the volume of a solid by rotating a region around an axis. Here's a detailed on how to use the washer method

It depends on the axis of rotation, the formula for the volume V using the washer method is: