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 December 28th, 2012, 06:11 PM #1 Newbie   Joined: Jun 2012 Posts: 15 Thanks: 0 When do I use the washer method? Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. y=x^2 , x=y^2; about y=1 Why is the area: A(x)=Pi[(1-x^2)^2-(1-sqrt(x))^2]? I know how to do the problem, but I want to know where the is 1 - X^2 coming from? Whis is there a 1 - in front? When Do I know when I have to use it?
 December 28th, 2012, 06:45 PM #2 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,155 Thanks: 463 Math Focus: Calculus/ODEs Re: When do I use the washer method? For an arbitrary washer, the outer radius is the distance between the axis of rotation and the curve $y=x^2$, while the inner radius is the distance between the axis of rotation and the curve $y=\sqrt{x}$. As far as when to use the washer method, do so if instructed, or if given a choice, when it is easier to compute than the shell method. I recommend using both methods though, to check your work, and for practice using either method.
 December 28th, 2012, 06:54 PM #3 Global Moderator     Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,642 Thanks: 960 Math Focus: Elementary mathematics and beyond Re: When do I use the washer method? I get $V\,=\,\frac{3\pi}{10}$.
December 28th, 2012, 06:54 PM   #4
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Re: When do I use the washer method?

Quote:
 Originally Posted by MarkFL For an arbitrary washer, the outer radius is the distance between the axis of rotation and the curve $y=x^2$, while the inner radius is the distance between the axis of rotation and the curve $y=\sqrt{x}$. As far as when to use the washer method, do so if instructed, or if given a choice, when it is easier to compute than the shell method. I recommend using both methods though, to check your work, and for practice using either method.
Yes but when do I know when to put 1- in front of the function in area? What should the question day then?

December 28th, 2012, 06:55 PM   #5
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Re: When do I use the washer method?

Quote:
 Originally Posted by greg1313 I get $V\,=\,\frac{3\pi}{10}$.

December 28th, 2012, 06:57 PM   #6
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Re: When do I use the washer method?

Quote:
 Originally Posted by MarkFL For an arbitrary washer, the outer radius is the distance between the axis of rotation and the curve $y=x^2$, while the inner radius is the distance between the axis of rotation and the curve $y=\sqrt{x}$. As far as when to use the washer method, do so if instructed, or if given a choice, when it is easier to compute than the shell method. I recommend using both methods though, to check your work, and for practice using either method.
When should I know when a cross section with a washer needs to have 1 - in front of it? That is my question.

 December 28th, 2012, 07:06 PM #7 Global Moderator     Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,642 Thanks: 960 Math Focus: Elementary mathematics and beyond Re: When do I use the washer method? $\pi\int_a^b$R_0(y)$^2\,-\,$R_1(y)]^2\,dy$ $\pi\int_0^1(\sqrt{y})^2\,-\,(y^2)^2\,dy\,=\,\pi\int_0^1y\,-\,y^4\,dy\,=\,\pi$$\frac12\,-\,\frac15$$\,=\,\frac{3\pi}{10}$ December 28th, 2012, 07:14 PM #8 Newbie Joined: Jun 2012 Posts: 15 Thanks: 0 Re: When do I use the washer method? Quote:  Originally Posted by greg1313 $\pi\int_a^b\[R_0(y)$^2\,-\,\[R_1(y)]^2\,dy$ $\pi\int_0^1(\sqrt{y})^2\,-\,(y^2)^2\,dy\,=\,\pi\int_0^1y\,-\,y^4\,dy\,=\,\pi$$\frac12\,-\,\frac15$$\,=\,\frac{3\pi}{10}$
No the answer should be 11/30(Pi)

 December 28th, 2012, 07:16 PM #9 Global Moderator     Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,642 Thanks: 960 Math Focus: Elementary mathematics and beyond Re: When do I use the washer method? Where did you get that answer?
December 28th, 2012, 07:20 PM   #10
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Re: When do I use the washer method?

Quote:
Originally Posted by mortified_penguin
Quote:
 Originally Posted by MarkFL For an arbitrary washer, the outer radius is the distance between the axis of rotation and the curve $y=x^2$, while the inner radius is the distance between the axis of rotation and the curve $y=\sqrt{x}$. As far as when to use the washer method, do so if instructed, or if given a choice, when it is easier to compute than the shell method. I recommend using both methods though, to check your work, and for practice using either method.
Yes but when do I know when to put 1- in front of the function in area? What should the question day then?
The distance between the axis of rotation (y = 1 in this case) and some function beneath it is 1 - f(x). The "1 -" comes from that distance.

I get $V=\frac{11\pi}{30}$.

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