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# Washer Method Calculator

To calculate volume of the solid revolution, enter **f(x)**, **g(x)**, and upper & lower limit into the required input fields and click the **calculate **button using washer calculator

## Washer Method Calculator

Washer method calculator finds the volume of the solid revolution to cover the sold with a hole by using a definite integral. This washer calculator finds the definite integral of the sum of two squared functions **(f(x) ^{2} + g(x)^{2})** and multiplies it by

**π (pi)**.

## What is the washer method?

In geometry, a washer method is used to find the volume of different kinds of solid shapes such as a round shape with a hole in the center. The shapes are obtained by rotating two functions around the **x-axis **and **y-axis.**

## Formula of washer method

The formula for the washer method is:

## How to calculate the problems of the washer method?

The washer method calculator above can solve the problems in a couple of seconds. Below is a solved example to calculate the washer method manually.

**Example**

Find the volume of the solid revolution if the functions are` f(x) = x`

& ^{2} + 4`g(x) = 4x + 1`

in the interval of **[2, 4]**.

**Solution **

**Step 1:** Take the given information.

f(x) = x^{2 }+ 4

g(x) = 4x + 1

upper limit = 4

lower limit = 2

**Step 2: **Take the **formula **of the washer method.

Washer method = π\(\int _a^b\:\)[f(x)2 + g(x)2] dx

**Step 3:** Calculate the sum of squares of the functions

f(x)^{2} = (x^{2} + 4)^{2} = x^{4} + 8x^{2} + 16

g(x)^{2} = (4x + 1)^{2} = 16x^{2} + 8x + 1

f(x)^{2 }+ g(x)^{2} = x^{4} + 8x^{2 }+ 16 + 16x^{2} + 8x + 1

`f(x)`

^{2} + g(x)^{2 }= x^{4} + 24x2 + 8x + 17

**Step 4: **Find the definite integral of the above function with respect to “**x**”.

\(\int _2^4\:\)[f(x)^{2} + g(x)^{2}] dx = \(\int _2^4\:\)[x^{4} + 24x^{2} + 8x + 17] dx

= \(\int _2^4\:\)[x^{4}] dx + \(\int _2^4\:\)[24x^{2}] dx + \(\int _2^4\:\)[8x] dx + \(\int _2^4\:\)[17] dx

= [x^{4}+1] dx + \(\int _2^4\:\)[24x^{2}] dx + \(\int _2^4\:\)[8x] dx + \(\int _2^4\:\)[17] dx

\(=\left[\frac{x^{4+1}}{4+1}\right]^4_2+24\left[\frac{x^{3+1}}{3+1}\right]^4_2+\:8\left[\frac{x^{1+1}}{1+1}\right]^4_2\:+\:17\left[x\right]^4_2\)

\( =\left[\frac{x^5}{5}\right]^4_2+24\left[\frac{x^4}{4}\right]^4_2+\:8\left[\frac{x^2}{2}\right]^4_2\:+\:17\left[x\right]^4_2\)

\( =\frac{1}{5}\left[x^5\right]^4_2+\frac{24}{4}\left[x^4\right]^4_2+\:\frac{8}{2}\left[x^2\right]^4_2+17\left[x\right]^4_2\)

\( =\frac{1}{5}\left[4^5-2^5\right]+6\left[4^4-2^4\right]+4\left[4^2-2^2\right]+17\left[4-2\right]\)

\( =\frac{1}{5}\left[1024-32\right]+6\left[256-16\right]+4\left[16-4\right]+17\left[4-2\right]\)

\( =\frac{1}{5}\left[992\right]+6\left[240\right]+4\left[12\right]+17\left[2\right]\)

\( =198.4+1440+48+34\)

\(\int _2^4\:\left[x^4+24x^2+8x\:+17\right]\:dx=1720.4\)

**Step 5: **Substitute the values in the formula.

Washer method = π\(\int _2^4\:\)[x^{4 }+ 24x^{2} + 8x + 17] dx

Washer method = π(1720.4)

**Put π = 3.14**

`Washer method = (3.14)(1720.4) = 5402.056`