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Washer Method Calculator + Online Solver With Free Easy Steps

The online Washer Method Calculator is an online calculator that helps you find the volume of a disc using the washer method.

The Washer Method Calculator is a powerful tool used by mathematicians, physicists, and scientists to solve complex problems.

What Is a Washer Method Calculator?

A Washer Method Calculator is an online tool that can calculate the volume of a disk or a washer using the washer method.

The Washer Method Calculator requires four inputs to work: the first function equation, the second function equation, the starting interval, and the ending interval.

After inputting these values, the Washer Method Calculator calculates the disk area using the washer method.

How To Use a Washer Method Calculator?

To use the Washer Method Calculator, you must simply input the values and click the “Submit” button.

The detailed step-by-step instructions on how to use a Washer Method Calculator are given below:

Step 1

In the first step, we add the first function f(x) to the Washer Method Calculator.

Step 2

After adding the first equation f(x) we enter the second function equation g(x) in our Washer Method Calculator.

Step 3

When we are done with both the functions, we enter the first interval value in the Washer Method Calculator.

Step 4

After adding the first interval value, we proceed to add the second interval value in our Washer Method Calculator.

Step 5

Once we enter all the inputs in their respective boxes, we click the “Submit” button on the Washer Method Calculator. The Washer Method Calculator calculates the volume of the disc and displays it in a new window.

How Does a Washer Method Calculator Work?

A Washer Method Calculator works by taking in all the inputs and applying the washer method to the equations. The general equation for a washer method is shown below:

\[ V = \pi\int_{a}^{b}(R^{2}-r^{2}) dx  \quad \]

where R = Outer Radius , r =Inner Radius 

The washer method equation can also be written as:

\[ V = \int_{a}^{b}(\pi{R^{2}}-\pi{r^{2}}) dx  \quad\]

where R = Outer Radius , r =Inner Radius 

What Is a Disk Method?

The disk method is a formula for integration that can determine the volume of specific solids. The solid is divided into small disks (cylinders) using the disk method, and the greater overall volume is estimated by adding the volumes of the disks. 

It is important to remember that anti-derivatives, which determine the area under curves by defining the limit of rectangular areas as the width of the rectangles approaches zero, are related to integrals.

A three-dimensional shape must be made of stacked circular cross-sections, which may have different radii throughout the length of the solid, to employ the disk method. Water bottles, fruit cans, and filled vases are a few examples of three-dimensional things that fit the needed structure.

You can use the disk method formula as a function of either x or y. If a curve is rotated about the x-axis or a horizontal line, the integral is typically written as a function of x. 

If a curve is being rotated about the y-axis or a vertical line, write the integral as a function of y. Before applying the disk method formula, rephrase the curve being rotated using the function if it is not expressed in terms of the correct variable.

The formulas for the disk method is shown below:

\[ V = \int_{a}^{b} \pi(r(x))^{2}dx = \pi \int_{a}^{b} r(x)^{2}dx \quad with \ respect \ to \ x \] 

\[ V = \int_{c}^{d} \pi(r(y))^{2}dy = \pi \int_{c}^{d} r(y)^{2}dy \quad with \ respect \ to \ y \]

What Is the Washer Method?

The washer method is a method used to calculate the volume enclosed between two functions. This technique divides the revolution region perpendicular to the revolution axis. We refer to it as the “Washer Method” since the slices produced in this manner resemble washers. This method extends the disk method for calculating the volume of hollow solids in revolutions.

In construction, a washer is a thin plate with a hole in the middle that is used to disperse weight under a bolt or screw. In mathematical terminology, a washer is a circle with a smaller circle inside of it. 

To calculate the area of this shape, first, calculate the area of the larger circle, then calculate the area of the smaller circle, and finally subtract the two areas.

To derive the washer method formula we let f(x) and g(x) be continuous functions in [a,b] that are non-negative and such that $g(x) \leq f(x)$. Let R1 be the area enclosed in [a,b] by the two functions f(x) and g(x). 

By rotating the region, R around the x-axis, a solid is created, and its volume is given by:

\[ V = \pi\int_{a}^{b}f(x)-g(x) dx \]

However, the area of the circle is $A = \pi r^{2}$ we can rewrite the washer method formula as:

\[ V = \pi\int_{a}^{b}(R^{2}-r^{2}) dx \quad\]

where R = Outer Radius , r =Inner Radius 

Solved Examples

The Washer Method Calculator quickly provides you with the volume of a disk. 

Here are some examples solved using the Washer Method Calculator:

Example 1

A college student needs to calculate the volume of a hollow cylinder. The student calculates the following values:

f(x) = 2x + 16 

g(x) = -4x + 3 

Intervals = [-3,3] 

Using the Washer Method Calculator, find the volume of the cylinder.

A college student needs to calculate the volume of a hollow cylinder. The student calculates the following values:

f(x) = 2x + 16 

g(x) = -4x + 3 

Intervals = [-3,3] 

Using the Washer Method Calculator, find the volume of the cylinder.

Solution

We use the Washer Method Calculator to find the cylinder’s volume instantly. First, we enter the first function into its respective box; the first equation is f(x) = 2x + 16. After entering the first function, we enter the second function in the Washer Method Calculator; the second function is -4x + 3.

After we have entered both functions in our calculator, we add the first interval value; the first interval value is -3. Next, we add the second interval value in the Washer Method Calculator; the second interval value is 3.

Once all the input values are inputted, we click the “Submit” button present on the Washer Method Calculator. The calculator computes the volume of the cylinder and displays it below the calculator.

The following results are extracted from the Washer Method Calculator:

Definite Integral:

\[ V = \pi\int_{-3}^{3}(-(3-4x)^{2}+(16+2)^{2})dx = 1266 \pi \approx 3977.3 \]

Indefinite Integral:

\[ V = \pi\int (-(3-4x)^{2}+(16+2x)^{2})dx = \pi (-4^{3}+44x^2+247x)+constant \]

Example 2

An archeologist needs to find the volume of an ancient vase. The archeologist measured the vase and derived the following equations:

f(x) = 6x-2 

g(x) = -3x + 10 

Interval [-2,4] 

Calculate the volume of the vase using the Washer Method Calculator.

Solution

Using the Washer Method Calculator, we can quickly calculate the volume of the vase. Initially, we input the first function into the Washer Method Calculator; the value of the first function is f(x) = 6x-2. After entering the first equation, we enter our second function equation into its respective box; the second function is g(x) = -3x + 10.

Once we have plugged in both the functions in the Washer Method Calculator, we type in the first interval value; the first interval value is -2. After entering the first interval value, we plug in the second interval value in our Washer Method Calculator; the second interval value is 4.

Finally, once all the input values are entered into the calculator, we click the “Submit” button on the Washer Method Calculator. The calculator instantly displays the volume of the vase below the Washer Method Calculator.

The following results are generated using the Washer Method Calculator:

Definite Integral:

\[V = \pi\int_{-2}^{4} (-(10-3x)^{2}+(-2+6x)^{2})dx = 288\pi \approx 904.78 \]

Indefinite Integral:

\[ V = \pi\int (-(10-3x)^{2}+(-2+6x)^{2})dx = 3\pi (3x^{3}+6x^{2}-32x)+constant \]

Example 3

A physicist needs to calculate the volume of an uneven tube. The physicist calculates the following equations:

f(x) = 5x + 24 

g(x) = -2x + 14 

Intervals = [-1,2] 

Using the Washer Method Calculator, find the volume of the tube.

Solution

We use the Washer Method Calculator to compute the tube volume easily. First, we plug in the first function given to us in the Washer Method Calculator; the first function is f(x) = 5x + 24. After adding the first function, we add the second function to the calculator; the second equation is g(x) = -2x + 14.

After we have inputted both the functions, we start entering the interval values in our calculator. We enter the first interval value in its respective box; the first interval value is -1. Similarly, we add the second interval value in our Washer Method Calculator; the second interval value is 2.

Now all the inputs have been entered into the Washer Method Calculator. We click the “Submit” button, which instantly displays the tube volume.

The following results are computed using the Washer Method Calculator:

Definite Integral:

\[ V = \pi\int_{-1}^{2} (-(14-2x)^{2}+(24+5x)^{2})dx = 1647 \pi \approx 5174.2 \]

Indefinite Integral:

\[ V = \pi\int(-(14-2x)^{2}+(24+5x)^{2})dx = \pi(7x^{3}+148x^{2}+380x) + constant \]

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