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Thread: Washer method "solid of revolution" problem

  1. #1
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    Washer method "solid of revolution" problem

    Problem with setting up Washer/Disk method for determining solid of revolution.

    Need to find the volume of the solid of revolution:


    Find the volume of the solid of revolution generated by revolving region about the line y=7

    y=7sqrtx, y=7 and x=0

    I set up the problem like this:

    a=0 b=1 pi[7^2 - (7sqrtx)^2 dx] using the washer method

    The answer to this problem is 49pi/6 in my book.
    I got 49pi/2.

    How am I setting up the problem wrong? These are difficult problems, and getting an answer does not help with the learning.

  2. #2
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    Cross sections are "disks", not "washers", because there is no hole- the figure has y= 7 as one boundary and rotation is around y= 7. A line segment, perpendicular to y= 7, ending at [tex]y= 7\sqrt{x}[/tex] has length [tex]7- 7\sqrt{x}= 7(1- \sqrt{x})[/tex]. So a disk at a given x will have radius [tex]7- 7\sqrt{x}= 7(1- \sqrt{x})[/tex] and area [tex]\pi(7- 7\sqrt{x})^2= 49\pi(1- \sqrt{x}))^2[/tex] so a disk with thickness [tex]\Delta x[/tex] has volume [tex]49\pi(1- \sqrt{x})^2\Delta x[/tex]. The volume is approximately the sum of those, [tex]49\pi\sum(1- \sqrt{x})^2\Delta x[/tex]. In the limit, as the thickness goes to 0 and the number of disks goes to infinity, is the integral [tex]49\pi\int_0^7 (1- \sqrt{x})^2 dx[/tex].
    Last edited by HallsofIvy; 02-14-2019 at 01:01 PM.

  3. #3
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    Quote Originally Posted by calc67x View Post
    Problem with setting up Washer/Disk method for determining solid of revolution.

    Need to find the volume of the solid of revolution:


    Find the volume of the solid of revolution generated by revolving region about the line y=7

    y=7sqrtx, y=7 and x=0

    I set up the problem like this:

    a=0 b=1 pi[7^2 - (7sqrtx)^2 dx] using the washer method

    The answer to this problem is 49pi/6 in my book.
    I got 49pi/2.

    How am I setting up the problem wrong? These are difficult problems, and getting an answer does not help with the learning.
    Professor Halls showed you how to do the problem with the disc method (because you used dx). Now if the problem said to use the washer method then here is how you would proceed.

    a=o to b=7 (2pi*r*x)dy = a=o to b=7 (2pi*(7-y)(y2/49)dy

    These are only difficult if you do not do lots of them. When they start all looking the same, then you are done studying. Until then, you need to practice more.
    A mathematician is a blind man in a dark room looking for a black cat which isnít there. - Charles R. Darwin

  4. #4
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    Thanks!

    Great, this is exactly what I was looking for!

    Quote Originally Posted by Jomo View Post
    Professor Halls showed you how to do the problem with the disc method (because you used dx). Now if the problem said to use the washer method then here is how you would proceed.

    a=o to b=7 (2pi*r*x)dy = a=o to b=7 (2pi*(7-y)(y2/49)dy

    These are only difficult if you do not do lots of them. When they start all looking the same, then you are done studying. Until then, you need to practice more.

  5. #5
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    Quote Originally Posted by calc67x View Post
    Great, this is exactly what I was looking for!
    HallsofIvy showed you the disk/washer method, which involves circular or annular slices perpendicular to the axis. Jomo's method is actually the cylindrical shell method, where strips parallel to the axis are revolved around the axis. So that is not really what you were asking for.

  6. #6
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    Quote Originally Posted by Dr.Peterson View Post
    HallsofIvy showed you the disk/washer method, which involves circular or annular slices perpendicular to the axis. Jomo's method is actually the cylindrical shell method, where strips parallel to the axis are revolved around the axis. So that is not really what you were asking for.
    Yes, of course that is the shell method. It's been so long that I forget the name. At least I did it correctly.
    A mathematician is a blind man in a dark room looking for a black cat which isnít there. - Charles R. Darwin

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