# How to Use the Washer Method to Find the Volume in Calculus?

• darewinder
In summary, the person is trying to find the volume of a region bounded by y=2x(2-x) and x-axis. They use the Shell Method to find the volume, and then they use the washer method to integrate the equation to find the area.
darewinder
Hi, I need help with this calc2 question. Basically they want me to find the volume obtained by region bounded by y=2x(2-x) and x-axis. and the line of rotation is x=4.

Basically i know that i need to set up an equation based on the x variable that will give me the area and then i need to run the integral from 0 to 8 but i have no clue how to do this.

I heard that there is also shell method and a precalcus way to solve this, any idea guys?

Thanks a bunch

I am also taking Calc 2, and we just so happen to be working on the same thing. You are correct about using the Shell method for this. You use the shell method for rotation around a vertical line.

This is the formula used:

$$V=\int_a^b 2\pi(Radius)(Height)\,dx.$$

In this case your shell height is just going to be: $$-2x^2+4x$$, and your shell radius should just be $$4-x$$. Your interval will actually only be from 0 to 4 because you're dealing with the radius not the diameter. So now you should have:

$$\begin{math}V = 2\pi\int_0^4 (4-x)(-2x^2+4x)\,dx = 2\pi\int_0^4 (2x^3-8x^2-4x+16)\,dx\end{math} \begin{math}= 2\pi\left[\frac{1}{2}x^4-\frac{8}{3}x^3-2x^2+16x\right]_0^4\end{math}$$

I just learned this stuff last week or so, so you'll want to double check.

Last edited:
yea you must be in my class, rutgers math 152? I think i know you.\

For the problem i know you can do it with the shell method, but they want us to do it in washer method. so i need to change the equations to set them as a variable of way so i can integrate with dy. and do you have any idea on how to do this problem using precalculus.

Thanks

The washer method is pretty simple, though it's not the best way to do this problem. The area of a washer is $$\pi(r_1^{}^2-r_1^{}^2)$$ where r1 is the radius of the outer circle, and r2 is the radius of the inner circle. Which gives us a standard function for the volume of solids where you are dealing with "washers":
$$V=\pi\int_a^b f(x)^2-g(x)^2\,dx$$ where f(x) is your upper function, and g(x) is your lower function.
Now the function $$y=2x(x-2)$$ is clearly a parabola. You say it is bound by the x axis. So we are really looking at the two points where it crosses the x-axis (0,0), and (0,2), and it's global maximum (1,2). In other words, if we integrate with respect to y, we will be integrating from 0 to 2. First you need to get the equation in terms of y which leaves two possibilities:
$$x=\frac{-4+\sqrt{16-8y}}{-4}$$
and
$$x=\frac{-4-\sqrt{16-8y}}{-4}$$
It is easier to rotate around the Y Axis, but obviously this does not produce the same results. We can modify the equations slightly so that the results are the same. $$x=\frac{-4+\sqrt{16-8y}}{-4}+2$$, and $$x=\frac{-4-\sqrt{16-8y}}{-4}+2$$. The former will be the lower function, and the latter will be the upper function. Now you can integrate:

$$V = \pi\int_0^2(\frac{-4-\sqrt{16-8y}}{-4}+2)^2-(\frac{-4+\sqrt{16-8y}}{-4}+2)^2\,dy$$

You will want to finish the integral, and check my work up to this point. It looks good to me.

## What is the "volume by washer method" used for?

The volume by washer method is a technique used in calculus to find the volume of a three-dimensional shape by rotating a two-dimensional shape around an axis. It is commonly used in problems involving solids of revolution.

## How does the "volume by washer method" differ from other volume calculation methods?

The volume by washer method differs from other methods, such as the volume by cross-sectional area method, in that it uses washers (or cylindrical shells) instead of slices to calculate the volume. This allows for a more accurate approximation of irregularly shaped objects.

## What are the steps involved in using the "volume by washer method"?

The first step is to draw a cross-section of the three-dimensional shape. Then, determine the inner and outer radii of the washers and the height of each washer. Next, set up the integral by using the formula V = π∫ (outer radius)² - (inner radius)² dx or V = π∫ (outer radius)² - (inner radius)² dy, depending on the axis of rotation. Finally, evaluate the integral to find the volume.

## What are some common mistakes made when using the "volume by washer method"?

One common mistake is using the wrong axis of rotation. It is important to carefully read the problem and determine the correct axis before setting up the integral. Another mistake is not taking into account the orientation of the shape and using the wrong formula for the washer radii. It is also important to accurately calculate the height of each washer.

## In what real-life situations is the "volume by washer method" applicable?

The volume by washer method can be used to calculate the volume of objects such as vases, bottles, and other curved shapes. It is also applicable in engineering and architecture for calculating the volume of structures with curved or irregular shapes. Additionally, it can be used in physics and chemistry to find the volume of objects with varying densities.

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