90% of the Disk Method process will be in the setup. The key to the process will always be defining your region, and finding our radius equation , R(x) . Having a good working knowledge of the standard family of graphs (i.e., linear, quadratic, cubic, square root, exponential, trig) will be required.
Step 1: Draw out the region that you are being asked to rotate.
Use the equation(s) you were given to draw out the region you are being asked to rotate .
This is where your knowledge of the standard families of graphs will be put to the test.
Step 2: Determine the axis you are begin asked to revolve or rotate your region around.
You will be told around which axis they would like to you to rotate your region. It will often times be the xaxis or the yaxis . However, the axis could also be another horizontal for vertical line entirely (i.e., y = 2 or x = 3 ).
The language often used to identify the axis is “about the …”
Step 3: Determine whether your equation is in terms of x (standard: $y={\textcolor[rgb]{}{x}}^{2}+3\textcolor[rgb]{}{x}+\sqrt{\textcolor[rgb]{}{x}}$) or in terms of y (nonstandard: $x={\textcolor[rgb]{}{y}}^{2}+3\textcolor[rgb]{}{y}+\sqrt{\textcolor[rgb]{}{y}}$).
The axis of revolution will also tell you whether you need to take your integral with respect to x , dx , or whether you need to take it with respect to y , dy .
Step 4: Determine the bounds of the integral, $\underset{\textcolor[rgb]{}{a}}{\overset{\textcolor[rgb]{}{b}}{\int}}$, for each of your enclosed regions.
Intersections as Bounds

Equations as Bounds

Step 5: Determine your radius equation, $\textcolor[rgb]{}{R}\textcolor[rgb]{}{\left(}\textcolor[rgb]{}{x}\textcolor[rgb]{}{\right)}$.
You may need to solve that equation so that it is in terms of x or in terms of y , but the given equation is the R(x) equation.
In these situations, you will need to apply the Top minus Bottom concept to create the R(x) equation.
You have a given equation and an axis of revolution equation . One of those equations is the Top and one of those equations is the Bottom. You will need to review your sketch to decide which is which.
Step 6: Setup and evaluate your Disk Methodintegral.
Use the results of Step 4 (your bounds ) and Step 5 (your R(x) equation) together using the Disk Methodintegral setup.
$\textcolor[rgb]{}{\mathit{Volume}}=\underset{\textcolor[rgb]{}{a}}{\overset{\textcolor[rgb]{}{b}}{\int}}\textcolor[rgb]{}{\pi}{\left[\textcolor[rgb]{}{R}\textcolor[rgb]{}{\left(}\textcolor[rgb]{}{x}\textcolor[rgb]{}{\right)}\right]}^{\textcolor[rgb]{}{2}}\mathit{dx}$