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anonymous
 5 years ago
Use the method of cylindrical shells to find the volume V generated by rotating the region bounded by the given curves about the yaxis.
y = x^2
y=0
x=1
anonymous
 5 years ago
Use the method of cylindrical shells to find the volume V generated by rotating the region bounded by the given curves about the yaxis. y = x^2 y=0 x=1

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amistre64
 5 years ago
Best ResponseYou've already chosen the best response.1HA!! shells, i knew they were comin

amistre64
 5 years ago
Best ResponseYou've already chosen the best response.1since we is going around the y axis; its simpler right now

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0yup :( i never used this method before ><

amistre64
 5 years ago
Best ResponseYou've already chosen the best response.1the radius of each shell moves from x=0 to x=1 right?

amistre64
 5 years ago
Best ResponseYou've already chosen the best response.1and the height of each shell is determined by: f(x)

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0wait, i know why its x=1, but why is it x=0

amistre64
 5 years ago
Best ResponseYou've already chosen the best response.1so the area of each shell, when flattened out is height* width...

amistre64
 5 years ago
Best ResponseYou've already chosen the best response.1umm... y axis means x=0 they are the same thing

amistre64
 5 years ago
Best ResponseYou've already chosen the best response.1the height = f(x) the width = 2pi [x] the area of any given shell = 2pi x[f(x)] so integrate that from [a,b] = [0,1] in this case

amistre64
 5 years ago
Best ResponseYou've already chosen the best response.12pi [S] x(x^2) dx ; [0,1] 2pi x^4  = (pi x^4)/4 = pi/4 4

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0how did you get rid of the 4th power in the numerator?

amistre64
 5 years ago
Best ResponseYou've already chosen the best response.1you dont... why would you?

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0but you put (pi x^4)/4 = pi/4 where did the 4th power go?

amistre64
 5 years ago
Best ResponseYou've already chosen the best response.1F(0) = 0 F(1) = pi/4 F(1)  F(0) = pi/4

amistre64
 5 years ago
Best ResponseYou've already chosen the best response.1i forgot i changed 2/4 to 1/2.... doh!

amistre64
 5 years ago
Best ResponseYou've already chosen the best response.1\[\frac{2\pi x^4}{4} = \frac{\pi x^4}{2}\]

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0much clearer, lol thank you :D

amistre64
 5 years ago
Best ResponseYou've already chosen the best response.1when im wrong; say im wrong...not ask me how im right lol. I just assume im right ;)

amistre64
 5 years ago
Best ResponseYou've already chosen the best response.1you understand the mechanics of the shell method tho?

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0all the methods like the washer, disk, etc all those methods seem so similiar to eachother and i get them all confused. because in the end youre always taking the antiderivative and plugging in the boundaries right?

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0like i understand theres certain formulas you have to follow for each method, which i have to memerize lol all of them seem really similiar

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0so for the shell method, basically youre taking half of it, and spreading it out which is why it turns to be a rectangle shape?

amistre64
 5 years ago
Best ResponseYou've already chosen the best response.1when we flatten out the "shell" we get a flat rectangular piece that the area is eay to determine right?

amistre64
 5 years ago
Best ResponseYou've already chosen the best response.1the width = 2pi x...can you tell me why?

amistre64
 5 years ago
Best ResponseYou've already chosen the best response.1we aint using half the shell, but the whole thing

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0i think it has to do with the radius, is that why the width is 2 pi x?

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0ohh is it the circumference of the cylnder?

amistre64
 5 years ago
Best ResponseYou've already chosen the best response.1it does, but i want to make sure you understand this :) its very basic and easy...exactly lol

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0lol i took a careful look at the cylinder and realized haha xD

amistre64
 5 years ago
Best ResponseYou've already chosen the best response.1the shell method makes things easier for some problems

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0thanks ;) so basically for the shell method we're plugging in f(x) into the formula and taking the antiderivative of it? then plug in the boundaries and solve?

amistre64
 5 years ago
Best ResponseYou've already chosen the best response.1lets think that thruough; we want to add up all the areas...which is what intagrating does; each area = 2pi x[f(x)]

amistre64
 5 years ago
Best ResponseYou've already chosen the best response.1{S} 2pi x[f(x)] dx is what we do right?

amistre64
 5 years ago
Best ResponseYou've already chosen the best response.1if there is a constant we can pull it aside right?

amistre64
 5 years ago
Best ResponseYou've already chosen the best response.12 pi goes out and what are we left with inside?

amistre64
 5 years ago
Best ResponseYou've already chosen the best response.1excatly; so we integrate 'x*f(x)'

amistre64
 5 years ago
Best ResponseYou've already chosen the best response.1do you se how that differes from your first statement of ; we int f(x)?

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0yesss, thanks for clearing that up for me :) this method seems a bit easier than the others lol

amistre64
 5 years ago
Best ResponseYou've already chosen the best response.1look at the disc method

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0oh, i have one question. when you solve for an integral, is it possible for it to be negative? or does it always have to be positive

amistre64
 5 years ago
Best ResponseYou've already chosen the best response.1the disc method take all the areas of a given circle; with a radius of f(x) and adds them up

amistre64
 5 years ago
Best ResponseYou've already chosen the best response.1volume is always a positive number

amistre64
 5 years ago
Best ResponseYou've already chosen the best response.1what is the formula for the the area of a circle ?

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0uh oh. darn i should of took the absolute value. i was so sure i did it right too. lol but thank you :)

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0and area of circle is pi r^2

amistre64
 5 years ago
Best ResponseYou've already chosen the best response.1so when we add up all the areas of the cicles thru integration we do: {S} pi [f(x)]^2 dx right? adding up the areas of circles

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0so f(x) is the radius?

amistre64
 5 years ago
Best ResponseYou've already chosen the best response.1look at the drawing i did in this one

amistre64
 5 years ago
Best ResponseYou've already chosen the best response.1and tell me what the radius of each given circle is?

amistre64
 5 years ago
Best ResponseYou've already chosen the best response.1that is all these volume of rotation problems amopunt to :)

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0i remember there was this one method i think where you had to divide by two, is there a method that involves that? or am i making stuff up lol :p

amistre64
 5 years ago
Best ResponseYou've already chosen the best response.1i dont think thats a volume of rotation one :) in some area propblems we are asked to find the area under a curve that hopes the y axis... and if the symmetry is the same from left to right side of the y axis; we get a skewed result

amistre64
 5 years ago
Best ResponseYou've already chosen the best response.1lets try something simple and see; suppose we want to find the area of a square that is sitting halfway between the sides

amistre64
 5 years ago
Best ResponseYou've already chosen the best response.1lets say 4 high and 4 wide; the area should be 16 right?

amistre64
 5 years ago
Best ResponseYou've already chosen the best response.1if we integrate f(x) = y = 4; from 2 to 2 we know we should get 16 right?

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0how come its from 2 and 2?

amistre64
 5 years ago
Best ResponseYou've already chosen the best response.1{S} 4 dx > 4x 4(2)  4(2) = 8  8 = 16, so that one doesnt apply to the /2 thing you discussed lol

amistre64
 5 years ago
Best ResponseYou've already chosen the best response.1the distance from 2 to 2 = 4 right?

amistre64
 5 years ago
Best ResponseYou've already chosen the best response.1and a square with sides = 4 would straddle the y axis would sit from 2 to 2 on the x axis right?

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0correct. i gotcha now :) lol

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0if the prob asked to rotate around the xaxis, its generally the same idea of solving it like we did with the y axis right?

amistre64
 5 years ago
Best ResponseYou've already chosen the best response.1i pic is worth a thousand words :)

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0it really is. thank you for that :D

amistre64
 5 years ago
Best ResponseYou've already chosen the best response.1yes, but we just need to make sure we are rotating it properly; make sure the numbers come out correctly

amistre64
 5 years ago
Best ResponseYou've already chosen the best response.1once we can draw a picture, the rest is just intuitive

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0gotcha. i shall try and attempt the rest of these shell method questions. thank you so so much for all your help. :D you make it possible for me to understand calculus lol xD thank you times infinity !
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