Rotations of Functions

So I am in BC Calc in my high school and we started rotations of functions. Now, the way i see it, a cylinder should be able to be evaluated to a rectangular prism, with r=l, and h=h, and then seeing as a full rotation is 2PI w=2PI. But it doesn't work for the rectangle. Can Anyone tell me where I've gone wrong?

Not unless you ask your question in a more comprehensible manner..

>>1

I can't quite understand what you're saying, but I think you're talking out the cylindrical shells method. Basically you find the volume of a rotation by adding up a bunch of "shells" stacked inside one another like russian dolls.  You'll learn about it in Calc II I think, maybe Calc I.

Of course your post made almost 0 sense, so I could be way off base here.

Ummm... wot?

Let's see:
calculus and rotations of functions
cylinder and rectangular prism
r = 1
h = h
2πw = 2π

???

Don't know what your saying but I'll go ahead and show you how to find the volume of a cylinder with:
calculus and rotations of functions
a rectangle
radius = 1
height = h

Firstly, the volume of integration of function f(x) between a and b rotated around the x-axis: V = π∫(y)²dx,a,b

OK now then. To find your volume i'm going to rotate f(x)=h around the y-axis between 0 and 1.

Here's a crappy diagram (WARNING CRAPPY DIAGRAM IS CRAPPY):
,,,,,,,___,,,,,,..
,,,,,--,[b]|[/b],--,,,,..
,,,,/,,,[b]|___[/b]\,,,<h
,,,,\,,,[b]|[/b],,,[b]|[/b],,,..
,,,,|--_[b]|[/b]_--[b]|[/b],,,..
,,,,|,,,[b]|[/b],,,[b]|[/b],,,..
,,,,|,,,[b]|[/b],,,[b]|[/b],,,..
,[b]--------------[/b],<0
,,,,\,,,|,,,/,,,..
,,,,,--___--,,,,..
,,,,,,,,,,,,,,,,..
........^...^...
........0...1...
The axis and rectangle we will rotate are bold.

Now the volume we want to find is (since this is around the y-axis we swap the y to x, dx to dy and the a and b values to 0 and h):
V = π∫(x)²dy,0,h

Since the x value is 1 throughout, x=1
V = π∫(1)²dy,0,h
V = π∫1dy,0,h
V = π[y],0,h
V = π(h - 0)
V = πh

Working out the volume of a cylinder with radius = 1 and height = h normally to make sure it's correct:
V = πr²h
V = π×1²h
V = π×1×h
V = πh

Have fun!

what the hell do prisms have to do with volumes?
A rectangular prism would have volume l*w*h, so if you say had a rectangular prism with b = f(x) and w = g(x), h = k(y), then you would evaluate your area as 1/2f(x)g(x), then integrate that for     \int_a^b k(y)\,dy.
Notice that your base is purely a function of x, and remains constant for k(y).

Similar for a cylinder, except you would use your area of base equation as that of a circle, or oval.

>>4
Blah screwy vbcode ruined my creation.

I know shell method...
If you cut the cylinder along the line that marks the radius, and then an infinitely small distance away you cut again, making a rectangular prism that is the (integral of the rectangle)*x as x->0. Now you do that around the cylinder and make an infinite number of infinitely small cuts. Or at least as n->infinity, now you stack the rectangles on top of each other. That means, you would have the original rectangle(l*h) with an unknown width, but seeing as a full rotation in radians is 2π you can assume that the infinitely small widths you cut from the cylinder=2π when combined.
So then. l*w*2π should equal the same area of the cylinder, but it doesn't.

"area of a cylinder"

lol 3d.

w00t?!

Volume, my bad... can anyone help me? it seems like it should make sense but i can't find whats wrong

>>7
The rectangles have non-empty intersections for any non-zero thickness, unlike shells or "pie slices." Consequently doing this will not give a correct answer.

1. the area under a graph is its equation intergrated
2. rotating a graph about an axis or line would make a 3D shape
V = PI  multiplied by  (F(x))^2 = profit