Method: Washer Method

90% of the Washer Method, like the Disk Method,will be the setup. The keys to the process will always be defining your region, and finding your two radius equations, the Outer radius , R(x) , and the Inner radius , 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 x-axis or the y-axis . 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: $y=x^2+3x+\sqrt{x}$) or in terms of y (non-standard: $x=y^2+3y+\sqrt{y}$).

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 .

• If you are given the axis of revolution as the x-axis or a y =__ (a horizontal axis ), then you would need your equation in terms of x (standard: $y=x^2+3x+\sqrt{x}$).
• If you are given the axis of revolution as the y-axis or a x=__ (a vertical axis ), then you would need your equation in terms of y (non-standard: $x=y^2+3y+\sqrt{y}$).

Step 4: Determine the bounds of the integral, $\underset{a}{\overset{b}{\int }}$, for each of your enclosed regions.

As you have seen with previous integral technique, the bounds sometimes be found by finding where your curves intersect, and sometimes those bounds are given in the form of some additional vertical lines (i.e., x = 5 ) or horizontal lines, (i.e., y = 3 ).

• Your bounds are either going to be the intersection of your two equations, or they will be defined by an equation you were given.

Intersections as Bounds

Equations as Bounds

 

• If you are given an actual graph to work from, then you should be able to just read the bounds of your integral straight off the graph.
• If you are given equations and allowed to use your calculator, then you will be able to calculate the intersects of your two graphs using the Calc–Intersect function on your TI-84 calculator.
• When the region is closed off (bounded) on the left and right side by additional given equations which are vertical lines (i.e., x=2 , x=5 ), then those vertical lines will create your bounds.
• Sometimes you will be given only the equations, you will not be allowed to use your calculator, and you will have to calculate the intersects of those two equations by setting the two equations equal to each other and solving for the intersects (solving for x ).

Step 5: Determine your radius equations, the Outer radius , R(x) , and the Inner radius , r(x) .

This is definitely the crux of the whole process. You need to determine what the two radius equations will be. When you look at the drawing of your region from Step 1 , the equation that is on Top (above the other equation) will be the Outer radius , R(x) . The equation that is on Bottom (below the other equation) will be the Inner radius , r(x) .

When you are first doing these problems, I find it helpful to draw the two radiuses. Start on the axis of rotation you are provided, and draw a line towards the region that you drew in Step 1 .

• If you are rotating strictly around the standard x-axis or the y-axis , then the graph that you run into first is your Inner radius , r(x) , and the graph that you run into second is your Outer radius , R(x) .
• If you are rotating around a nonstandard axis , perhaps y = 6 or x = 4 , then you will need to create a new equation for the Outer radius , R(x) , and the Inner radius , r(x) .

In these situations, you have a given equation and an axis of revolution equation . You will need to apply the Top minus Bottom technique to create an equation that provides the distance between your given equation and the axis of revolution . One of those equations is the Top (above the other)and one of those equations is the Bottom (below the other). You will need to review your sketch to decide which is which.

Step 6: Setup and evaluate your Washer Methodintegral.

Put the results of Step 4 and Step 5 together using the Washer Method integral setup.

$V\mathit{olume}=\underset{a}{\overset{b}{\int }}\pi \left({\left[R\left(x\right)\right]}^2–{\left[r\left(x\right)\right]}^2\right)\mathit{dx}=\underset{a}{\overset{b}{\int }}\pi \left({\left[\mathit{Outter}\right]}^2–{\left[\mathit{Inner}\right]}^2\right)\mathit{dx}$