How Do You Know When to Use the Disk or Washer Method
AP Calculus Review: Disk and Washer Methods
The disk and washer methods are useful for finding volumes of solids of revolution. In this commodity, nosotros'll review the methods and work out a number of instance issues. By the end, yous'll be prepared for any disk and washer methods problems you lot encounter on the AP Calculus AB/BC exam!
Solids of Revolution
The disk and washer methods are specialized tools for finding volumes of certain kinds of solids — solids of revolution. So what is a solid of revolution?
Starting with a flat region of the airplane, generate the solid that would exist "swept out" as that region revolves around a fixed axis.
For example, if yous commencement with a right triangle, and and so revolve it around a vertical axis through its upright leg, then you get a cone.
The cone generated as a solid of revolution by revolving a right triangle effectually a vertical centrality
Here's another cool instance of a solid of revolution that you might take seen hanging up as a decoration! Tissue newspaper decorations that unfold from flat to round are examples of solids of revolution. Picket the next few seconds of the video below to see how it unfolds in existent time.
The Disk and Washer Methods: Formulas
So at present that you lot know a bit more about solids of revolution, permit'due south talk about their volumes.
Suppose South is a solid of revolution generated by a region R in the plane. At that place are two related formulas, depending on how complicated the region R is.
Disk Method
The simplest case is when R is the area nether a curve y = f(ten) between 10 = a and x = b, revolved around the 10-axis.
At present imagine cutting the solid into thin slices perpendicular to the x-centrality. Each piece looks similar a disk or cylinder, except that the outer surface of the disk may have a curve or slant. Permit's approximate each slice by a cylinder of height dx, where dx is very small.
In fact, I like to retrieve of each deejay as being generated past revolving a thin rectangle around the x-axis. And so you tin can see that the height of the rectangle, y, is the aforementioned as the radius of the deejay.
Now let'southward compute the book of a typical deejay located at position 10. The radius is y, which itself is just the function value at x. That is, r = y = f(x). The meridian of the disk is equal to dx (recollect of the disk every bit a cylinder standing on border).
Therefore, the volume of a single cylindrical disk is: V = π r two h = π f(10)2 dx.
This calculation gives the guess volume of a sparse slice of S. Side by side, to approximate the volume of the entirety of Due south, we have to add together up all of the disk volumes throughout the solid. For simplicity, assume that the thickness of each slice is constant (dx). Too, for technical reasons, we have to go on runway of the various ten-values forth the interval from a to b using the annotation xgrand for a "generic" sample point.
Finally, past letting the number of slices go to infinity (by taking a limit as n → ∞), we develop a useful formula for book every bit an integral.
Example 1: Deejay Method
Let R exist the region under the curve y = 2x three/2 between ten = 0 and x = 4. Detect the volume of the solid of revolution generated by revolving R around the 10-axis.
Solution
Let'south set the deejay method for this problem.
The book of the solid is 256π (roughly 804.25) cubic units.
Washer Method
Now suppose the generating region R is divisional by 2 functions, y = f(x) on the tiptop and y = 1000(ten) on the bottom.
This time, when you lot revolve R around an axis, the slices perpendicular to that axis will look similar washers.
No, we're not talking about clothes washers or dishwashers…
A washer is similar a deejay just with a center hole cut out. The formula for the volume of a washer requires both an inner radius r ane and outer radius r 2.
Nosotros'll need to know the volume formula for a single washer.
V = π (r two two – r ane 2) h = π (f(10)2 – k(x)2) dx.
Equally before, the exact volume formula arises from taking the limit as the number of slices becomes infinite.
Case 2: Washer Method
Decide the volume of the solid. Hither, the bounding curves for the generating region are outlined in ruby-red. The top curve is y = x and the bottom i is y = ten 2
Solution
This is definitely a solid of revolution. We'll set up the formula with f(10) = x (top) and g(x) = x 2 (lesser). Just what should we employ as a and b?
Well, just as in some expanse issues, yous may have to solve for the premises. Conspicuously the region is bounded by the two curves between their mutual intersection points. Set f(x) equal to g(ten) and solve to locate these points of intersection.
x = ten 2 → ten – x 2 = 0 → 10(1 – ten) = 0.
Nosotros find 2 such points: ten = 0 and one. And so prepare a = 0 and b = 1 in the formula.
Example iii: Different Axes
Fix an integral that computes the volume of the solid generated past revolving he region bounded past the curves y = 10 2 and ten = y iii effectually the line x = -one.
Solution
Be careful not to blindly use the formula without analyzing the situation offset!
This time, the centrality of rotation is a vertical line ten = -1 (rather than the horizontal x-axis). The radii will be horizontal segments, so retrieve of x 1 and x 2 (rather than y-values).
Furthermore, because everything is turned on its side compared to previous problems, we have to make certain both boundary functions are solved for 10. The thickness of the washer is now dy (instead of dx).
Finally, because the axis of revolution is 1 unit to the left of the y-axis, that adds another unit to each radius. (The further abroad the centrality, the longer the radius must be to reach the figure, correct?) Have a look at the graph below to assistance visualize what's going on.
- Inner Radius: x = y 3 + ane
- Outer Radius: ten = y i/2 + 1
As earlier, set the functions equal and solve for points of intersection. Those are again at x = 0 and 1.
Using the Washer Method formula for book, we obtain:
The problem but asks for setup, and so nosotros are done at this point.
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