Disk Method
Fully Explained w/ Step-by-Step Examples!

So what’s the disk method?

Jenn, Founder Calcworkshop^®, 15+ Years Experience (Licensed & Certified Teacher)

And how does it compare the shell or washer method?

All great questions, that we’re going to answer in today’s lesson.

Let’s jump in!

Background

What if I said that a cookie is a perfect way to visualize how to find the volume of a solid of revolution?

Yep!

Our goal is to take a two-dimensional bounded region and rotate it about a given axis to create a solid of revolution.

But how does this relate to a cookie?

First, we start with a rectangle because it’s a really easy object to find the area.

Axis Of Revolution For A Rectangle

Then, we spin this rectangle around a line and notice that it creates a circular shape.

Disk Method Around X Axis

Next, we realize that this rectangle creates more than just a circle because it has a thickness (width). And if this circle has a thickness, it makes something that looks like a disk.

And a disk looks just like a cookie! Gosh, math is delicious!

Disk Method Equations

Okay, now here’s the cool part.

We find the volume of this disk (ahem, cookie) using our formula from geometry:

\begin{equation}
\begin{array}{l}
V=(\text { area of base })(\text { width }) \\
V=\left(\pi R^{2}\right)(w)
\end{array}
\end{equation}

But this will only give us the volume of one disk (cookie), so we’ll use integration to find the volume of an infinite number of circular cross-sections of the 3D solid.

Disk Method Formulas

Cool!

Important Notes

Now here some critical things to note.

• First, our bounded region must be entirely flush against the axis of rotation to ensure that we will create a disk when rotated.
• Secondly, our green rectangle represents either a vertical or horizontal slice and will always be perpendicular to the axis of rotation.
• And lastly, our R-value represents the height of our rectangle, which is precisely the same as finding the area between two curves:

\begin{equation}
R=(\text { top function })-(\text { bottom function })
\end{equation}

Disk Method Example

Alright, so now let’s walk through an example to help us make sense of everything.

Suppose we are asked to find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the x-axis:

\begin{equation}
y=x, y=0, x=0 \text { and } x=4
\end{equation}

Step 1:

First, we will graph our bounded region.

Volume Of A Triangle Rotated About An Axis

Step 2:

Next, we will identify our axis of rotation and create our vertical, rectangular slice perpendicular to the axis of rotation (i.e., the x-axis).

Vertical Rectangular Cross Section Representation

Step 3:

Now we must determine our bounds, or width of the bounded region, which will become our bounds of integration as well as our top and bottom function.

Plane Region Disk Revolved About X Axis

Step 4:

Lastly, we plug everything into our formula and integrate it to find the volume of the resulting solid of revolution.

\begin{equation}
\begin{array}{l}
V=\pi \int_{a}^{b}\left(R^{2}\right) d x=\pi \int_{0}^{4}(x-0)^{2} d x \\
V=\pi \int_{0}^{4} x^{2} d x=\pi\left(\frac{x^{3}}{3}\right]_{0}^{4}=\pi\left(\frac{(4)^{3}}{3}-\frac{(0)^{3}}{3}\right)=\frac{64 \pi}{3}
\end{array}
\end{equation}

Wow! We just found that the volume of the bounded region when rotated about the x-axis!

Solid Of Revolution Disk Method

Summary

Together, we will work through numerous questions, step by step, to find the volume of a solid generated about the x-axis, y-axis, or any horizontal or vertical line, whose cross-sections are disks.

Now it’s time to grab a cookie (disk) and click on the video!

Video Tutorial w/ Full Lesson & Detailed Examples (Video)

Get access to all the courses and over 450 HD videos with your subscription

Monthly and Yearly Plans Available

Get My Subscription Now

Still wondering if CalcWorkshop is right for you?
Take a Tour and find out how a membership can take the struggle out of learning math.