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UFO Waterproof Remote found floating in the company of the rich and famous (Engadget)

Dreams are nice. Yachts are better. So while you cling to far-flung notions
that you'll ever truly know what 'money to burn' means, the Barry Dillers of
our world are crisscrossing international waters, drenching models with
diamonds and dropping their universal controllers in the hot tub with abandon.
Rescuing these rapscallion, modern-day robber barons from the pitfalls of
excess is Crestron, with its UFO Waterproof Remote made specifically for H2O
hanky panky. The disc-shaped unit, which admittedly looks more like a pool toy
than high-end control hub, comes encased in a rubberized shell and features a
2.8-inch display, giving Greek shipping heirs and their ilk instant access to
lighting, security systems, A/V equipment and thermostats from the comforts of
their sun deck jacuzzi. Alright, so the 1% aren't the company's sole market
base, as the unit's also ideally positioned for use in hospitals, owing to its
ability for easy sterilization. But let's be honest, the real reason this
floating controller shares a frisbee-like shape is aerodynamics. All the
better to hit your staff with, _right_ Ms. Campbell? Official PR and its
hydrophobic emphasis after the break.

Continue reading _UFO Waterproof Remote found floating in the company ...

Engadget

Solids of Revolution - Calculus?!?

Can comone please explain the difference between the shell, disk, and washer methods when finding volumes of solids of revolution?! I always get stumped when trying to figure out which one to use. Thanks for any help!

Disk
Function of x
If the function to be revolved is a function of x, the following integral represents the volume of the solid of revolution:

where R(x) is the distance between the function and the axis of rotation. This works only if the axis of rotation is horizontal (example: y = 3 or some other constant).

Function of y
If the function to be revolved is a function of y, the following integral will obtain the volume of the solid of revolution:

where R(y) is the distance between the function and the axis of rotation. This works only if the axis of rotation is vertical (example: x = 4 or some other constant).

Washer

To obtain a "hollow" solid of revolution (sometimes called the "washer method"), the procedure would be to take the volume of the inner solid of revolution and subtract from it the volume of the outer solid of revolution. This can be calculated in a single integral similar to the following:

Where RO(x) is the function that is farthest from the axis of rotation and RI(x) is the function that is closest to the axis of rotation. One should take caution not to evaluate the square of the difference of the two functions, but to evaluate the difference of the squares of the two functions.

NOTE: the above formula only works for revolutions about the x-axis.

To rotate about any horizontal axis, simply subtract from that axis each formula:

if h is the value of a horizontal axis, then the volume =

For example, to rotate the region between y = − 2x + x2 and y = x

along the axis y = 4, you would have to integrate as follows:

Note that when you integrate along an axis other than the x, the further axis may not be that obvious. In the previous example, even though y = x is further up than y = − 2x + x2, it is the inner axis since it is closer to y = 4

The same idea can be applied to both the y-axis and any other vertical axis. You simply must solve each equation for x before you plug them into the integration formula.

shell:
Mathematically, this method is represented by:

if the rotation is around the x-axis (horizontal axis of revolution), or

if the rotation is around the y-axis (vertical axis of revolution).

So here the function p(x) is the distance from the axis and h(x) is the length of the shell, generally the function being rotated. The values for a and b are the limits of integration, the starting and stopping points of the rotated shape (note the limits are units of the Axis of Revolution).