WEBVTT
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do you find the volume of the solid that is
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rotated of the region, bounded by why equals two
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minus 1/2 X y equals zero X equals one X
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equals two and we're going around the x axis.
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So first of all, let's consider that linear function
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to minus 1/2 X. This has I intercept of
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two and then a slope of negative 1/2 so down
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1/2 down, 1/2 would make this line. It's
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about a straight. I could draw it and then
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we're also looking at y equals zero. So that's
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gonna be your X axis. We're looking at X
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is wine, and we're looking at X is too
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. So the region that's bounded by that linear function
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X is one excess to And why zeros this part
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of shaded in green Now for rotating it around the
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X axis, Then I always want to ask myself
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, Is there any volume there any space there that
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needs to be cut out where that's missing? And
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the answer here is know, as we rotate this
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, it's gonna be a totally solid figure. I
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know this because every part of that shaded region and
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green is coming right up against the exact access.
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So because it's that solid figure, I know for
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sure it's a disk method. So for the disk
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method for volume, we have high R squared and
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we want to look for the ex boundaries. We
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want to have it in terms of X with our
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in a girl because we're going around the X axis
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. Thankfully, we're given the limits for X.
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The smallest X value is one the greatest Xlu is
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, too. That's what's bounding this volume. So
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I'm gonna put one into on my integral and then
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to fill in our you're thinking of the Radius.
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So if I'm looking at my figure, the radius
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is how far I travel from the center. So
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where I'm spinning Teoh, you know that that edge
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of the shape and that are here is determined by
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going from the linear function to minus 1/2 X down
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to the axis. So I always like to think
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of if I'm in terms of X is I like
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to think of it as upper minus lower. The
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lower function here, though, is just zero.
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Because it's the Axis, it's helpful to think upper
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minus lower for if I have shifted this somewhere else
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into the plane, so we're gonna fill into minus
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1/2 X as that are term here. So then
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we want to calculate this in a rural. It's
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pretty easy to go through by hand. If you
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would just expand that by no meal. So if
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you just multiply and then you can just do each
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term by term, so multiplying that out and then
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going term by term Or if you have a calculation
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program, you can find this integral total. The
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integral here is 19/12 so I wouldn't leave my answer
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exact and call it 19 pi over 12.