WEBVTT
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this problem here. What we're gonna want to do
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is we're going to want to be looking at,
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um, determining the volume of a solid, um
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, by rotating it around an axis. So the
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first thing you want to do is sketch what this
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solves gonna look like who always see that the line
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we're focusing on is X y equals X plus one
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. So I have a point here, and then
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what we'll have is a ruler. Um, look
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, something like this. Perhaps that would be our
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graph. And then we want Thio create a line
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like this now that we have this in place,
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um, we want to also sketch other lines that
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we have. We know that why goes from we
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have the line y equals zero right here. We
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also have X equals two. So 12 And that's
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really what we're gonna have for our solid this region
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right here is what we're gonna have. And then
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what we're gonna do is we're going to wrap it
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around the x axis. So when we wrap it
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around the X axis right here, we have our
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solid. We'll do some bloom. So this right
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here, we're gonna wrap it around the X axis
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. So it's gonna go like this and what we're
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gonna end up getting if you can picture this is
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a solid almost like a a cone shape or like
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a horn shape. It's gonna look something like that
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. And what we're gonna have is this is not
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gonna be a washer. It is going to be
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a disk. So now we want to do is
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find the volume of it. So the volume is
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going to be equal to pi times the integral from
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0 to 2 of the radius squared DX. We
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know the radius is going to be determined by X
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plus one, because if we're right here, it's
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X plus one away from the X axis. If
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we're here, it's X plus one away from the
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X axis. So with that in mind, we
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now want to replace the word radius with X plus
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one squared. Now that we have that, we
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see that V. The volume is equal to pi
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times the integral of, um, I find it
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easier to foil this out so we'll just foil it
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out to be plus one DX. Then we're going
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to differentiate this. What will end up getting is
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pie times. You'll get X cubed over three plus
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X squared, plus X evaluated at zero into.
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When we do this, we're gonna end up getting
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that. The volume is equal to pi times,
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um, 27 3rd minus one third. That's gonna
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be 26 3rd. So our volume is 26 pi
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over three, and that will be the final answer
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.