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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 t he 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-ais 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) = x2 + 4 & g(x) = 4x + 1 in the interval of [2, 4].
Solution
Step 1: Take the given information.
f(x) = x2 + 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 = (x2 + 4)2 = x4 + 8x2 + 16
g(x)2 = (4x + 1)2 = 16x2 + 8x + 1
f(x)2 + g(x)2 = x4 + 8x2 + 16 + 16x2 + 8x + 1
f(x)2 + g(x)2 = x4 + 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 = \(\int _2^4\:\)[x4 + 24x2 + 8x + 17] dx
= \(\int _2^4\:\)[x4] dx + \(\int _2^4\:\)[24x2] dx + \(\int _2^4\:\)[8x] dx + \(\int _2^4\:\)[17] dx
= [x4+1] dx + \(\int _2^4\:\)[24x2] 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\(\int _2^4\:\)[x4 + 24x2 + 8x + 17] dx
Washer method = π(1720.4)
Put π = 3.14
Washer method = (3.14)(1720.4) = 5402.056
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