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Washer method rotating around non-axis

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    Now let's do a really
    interesting problem.
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    So I have y equals x,
    and y is equal to x
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    squared minus 2x
    right over here.
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    And we're going to
    rotate the region
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    in between these two functions.
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    So that's this region
    right over here.
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    And we're not going to rotate
    it just around the x-axis,
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    we're going to rotate it around
    the horizontal line y equals 4.
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    So we're going to
    rotate it around this.
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    And if we do that, we'll get
    a shape that looks like this.
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    I drew it ahead of time, just
    so I could draw it nicely.
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    And as you can see, it looks
    like some type of a vase
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    with a hole at the bottom.
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    And so what we're going to do
    is attempt to do this using,
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    I guess you'd call it
    the washer method which
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    is a variant of the disk method.
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    So let's construct a washer.
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    So let's look at a given x.
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    So let's say an x
    right over here.
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    So let's say that we're
    at an x right over there.
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    And what we're
    going to do is we're
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    going to rotate this region.
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    We're going to give
    it some depth, dx.
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    So that is dx.
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    We're going to rotate
    this around the line y
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    is equal to 4.
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    So if you were to visualize it
    over here, you have some depth.
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    And when you rotate it
    around, the inner radius
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    is going to look like the
    inner radius of our washer.
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    It's going to look
    something like that.
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    And then the outer
    radius of our washer
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    is going to contour
    around x squared minus 2x.
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    So it's going to
    look something--
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    my best attempt
    to draw it-- it's
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    going to look
    something like that.
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    And of course, our washer
    is going to have some depth.
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    So let me draw the depth.
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    So it's going to
    have some depth, dx.
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    So this is my best attempt at
    drawing some of that the depth.
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    So this is the
    depth of our washer.
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    And then just to make the face
    of the washer a little bit
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    clearer, let me do it
    in this green color.
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    So the face of the
    washer is going
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    to be all of this business.
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    All of this business is going
    to be the face of our washer.
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    So if we can figure
    out the volume of one
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    of these washers for a
    given x, then we just
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    have to sum up all of
    the washers for all
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    of the x's in our interval.
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    So let's see if we can
    set up the integral,
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    and maybe in the
    next video we'll
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    just forge ahead and actually
    evaluate the integral.
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    So let's think about the
    volume of the washer.
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    To think about the
    volume of the washer,
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    we really just have to
    think about the area
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    of the face of the washer.
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    So area of "face"--
    put face in quotes--
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    is going to be equal to what?
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    Well, it would be the
    area of the washer--
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    if it wasn't a washer,
    if it was just a coin--
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    and then subtract out
    the area of the part
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    that you're cutting out.
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    So the area of the
    washer if it didn't
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    have a hole in the
    middle would just
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    be pi times the
    outer radius squared.
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    It would be pi times
    this radius squared,
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    that we could call
    the outer radius.
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    And since it's a washer,
    we need to subtract out
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    the area of this inner circle.
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    So minus pi times
    inner radius squared.
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    So we really just
    have to figure out
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    what the outer and inner radius,
    or radii I should say, are.
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    So let's think about it.
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    So our outer radius is
    going to be equal to what?
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    Well, we can visualize
    it over here.
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    This is our outer
    radius, which is also
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    going to be equal to
    that right over there.
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    So that's the distance
    between y equals
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    4 and the function that's
    defining our outside.
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    So this is essentially,
    this height right over here,
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    is going to be equal to 4
    minus x squared minus 2x.
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    I'm just finding the distance
    or the height between these two
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    functions.
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    So the outer radius is
    going to be 4 minus this,
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    minus x squared minus
    2x, which is just 4
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    minus x squared plus 2x.
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    Now, what is the inner radius?
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    What is that going to be?
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    Well, that's just going to
    be this distance between y
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    equals 4 and y equals x.
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    So it's just going
    to be 4 minus x.
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    So if we wanted to find
    the area of the face of one
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    of these washers for a
    given x, it's going to be--
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    and we can factor
    out this pi-- it's
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    going to be pi times the
    outer radius squared,
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    which is all of this
    business squared.
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    So it's going to be 4 minus
    x squared plus 2x squared
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    minus pi times
    the inner radius--
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    although we factored
    out the pi-- so
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    minus the inner radius squared.
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    So minus 4 minus x squared.
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    So this will give us
    the area of the surface
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    or the face of one
    of these washers.
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    If we want the volume of
    one of those watchers,
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    we then just have to
    multiply times the depth, dx.
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    And then if we want to actually
    find the volume of this entire
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    figure, then we just have to
    sum up all of these washers
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    for each of our x's.
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    So let's do that.
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    So we're going to sum
    up the washers for each
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    of our x's and take the
    limit as they approach zero,
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    but we have to make sure
    we got our interval right.
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    So what are these-- we care
    about the entire region
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    between the points
    where they intersect.
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    So let's make sure
    we get our interval.
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    So to figure out our
    interval, we just
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    say when does y equal
    x intersect y equal
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    x squared minus 2x?
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    Let me do this in
    a different color.
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    We just have to
    think about when does
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    x equal x squared minus 2x.
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    When are our two functions
    equal to each other?
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    Which is equivalent
    to-- if we just
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    subtract x from
    both sides, we get
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    when does x squared
    minus 3x equal 0.
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    We can factor out an x
    on the right hand side.
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    So this is going to be when does
    x times x minus 3 equal zero.
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    Well, if the product is equal
    to 0, at least one of these
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    need to be equal to 0.
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    So x could be equal to 0,
    or x minus 3 is equal to 0.
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    So x is equal to 0
    or x is equal to 3.
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    So this is x is 0, and
    this right over here
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    is x is equal to 3.
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    So that gives us our interval.
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    We're going to go
    from x equals 0
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    to x equals 3 to get our volume.
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    In the next video,
    we'll actually
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    evaluate this integral.
Title:
Washer method rotating around non-axis
Description:

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Video Language:
English
Team:
Khan Academy
Duration:
06:37

English subtitles

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