User talk:Santina0424

What are Washer and Shell?
The way to find a uniformly symmetric solid using calculus is not described in your Calculus book. However, it is one important aspect in BCII and it’s especially useful in real life. BCII course covers two methods in finding the volume of a solid; they are washer and shell

Brief Description
Washer (or disk) is a method of using the length of x over an infinitely small y interval (or vice versa) as the radius and sum up the volume of all the infinitely thin circular slices. A general form of washer method:

Where f(x) is the top function and g(x) is the bottom one on the interval (a, b) and the solid is obtained by revolving the area enclosed by these functions around x-axis. For convenience, we would define a bottom function as the function that is closer to the axis of revolution than the other function (top function).

Shell, on the other hand, is a method of summing up all the cylinders with infinitely small thickness by taking the radius and height into account. A general form of a shell method: , or

Where f(x) is the top function and g(x) is the bottom one on the interval (a, b) and the solid is obtained by revolving the area enclosed by these functions around y-axis. As you could see from the general form, the volume of each cylinder is obtained by subtracting the volume of cylinder created by g(x) from the one created by f(x).

A sample problem
Using either shell or washer method, find the volume created by revolving the area enclosed by and  around y-axis.

First step, sketch both functions and find the bounded region. As shown in the figure on the right, the region bounded by these two functions is on  [0, 1] and  [0, 1] The bottom function is, and the top function is.

Using shell methods, we could add the volume of all the hallow cylinders by subtracting the cylinders created by the bottom function from the ones created by the top function. The integral, followed the general formula, would be:

Or If you were to use washer, be sure to write each function in terms of y. Since we are adding the circular slices created by the two functions in respect to their length in x, we need to rewrite each function in terms of y.

Volume =

How do you know which method to used? In fact, both methods work out perfectly. However, you should always be caution the variable you are differentiating in respect to. Besides using your logical sense, Table 1 is something you could always keep in mind when you get blanked out during an assessment.

Revolve around a horizontal axis 	Revolve around a vertical axis Washer	dx	dy Shell	dy	dx

Sometimes you might encounter problems that require you to find the volume created by revolving a bounded area around a horizontal/vertical line that is neither x-axis nor y-axis. For this kind of problems, you would need to do some reasoning to take the shifting of the axis of revolution into account. Look closely to Figure 5 and then see if Table 2 makes sense to you.

Summary of shell and washer methods   t (top function)   b (bottom function)

Again, for convenience (at least in this wiki page), we would define a bottom function as the function that is closer to the axis of revolution than the other function (top function).

Around horizontal axis y=0	y=5	y= -2 Washer

Shell	2

Around vertical axis x=0	x=5	x= -2 Washer

Shell	2

Some reminders and tips 1.	It’s always whichever is furthest form the axis of revolution subtracts the one that’s closer to the axis. 2.	The bound of your integral should correspond to the variable you are using for your integrals.

extra links
Watch animations here: http://mathdemos.gcsu.edu/mathdemos/shellmethod/

More practice problems on this topic: http://learning.mgccc.cc.ms.us/math/mathdocs/calc/revol2.pdf