WEBVTT
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it's were given a region founded by curves and a
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line and were asked to set up an integral for
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the volume of the solid obtained by rotating this region
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bounded by these curves about this line Then rescues are
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calculated to evaluate the integral correct to five decimal places
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. The curves are y equals x squared and x
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squared plus y squared equals one. Oh yes where
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why is greater than or equal to zero? This
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is going to be The Parabola and the upper half
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of a circle of Radius one. And in part
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a the line is about the X axis like that
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. Mhm. Black Cat December 22 Central we were
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playing HFS's Yeah we're putting up opening for jimmy's chicken
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sack, Coachella. And so we're essentially rotating a
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thin vertical strip. Do A D. C.
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Did part about the X axis. We get a
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washer in other words with inner radius various X squared
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and an outer radius. Yeah. Well if you
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solve the equation for the circle and take why to
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be positive, this is the positive square root of
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one minus X squared. Oh, your will and
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grace. So we use the washer method and the
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volume V is pi times the integral from uh the
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okay, X coordinate of the intersection of Y equals
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x squared and x squared plus Y squared equals one
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. We'll we'll do this first actually. So when
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are these intersecting? Well, this is when Y
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plus. Okay. Just taking chest john great equals
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one. So solving for why we get Mhm.
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What? Well, we really don't want to find
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why we want to find X. So instead,
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yeah, we have X squared plus X squared squared
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equals one. So we have X to the fourth
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plus X squared minus one equals zero. And solving
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we get that X squared is equal to negative one
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plus or minus the square root of one plus four
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. All over to This is negative one Plus or
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-15/2. However, X squared is positive. Seriously
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considering one plus Route five you too and therefore X
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is going to be plus or minus the square root
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of route five minus 1/2. And if you have
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to calculate this is approximately point 786 Yeah, that
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fits Okay, so our volume V is pi times
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the integral from our left point of intersection which is
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negative square root of route five minus 1/2 2.
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Positive square root of route five minus 1/2. The
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of our outer radius squared. Mhm. This is
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the square root of one minus X squared minus our
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inner radius X squared squared d x£5000. And
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so simplifying this is pi times the integral from negative
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square root of route five minus 1/2 Square root of
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route 5-1/2 of this is one minus X squared
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. Mhm. Yeah. They all do that voice
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. Yeah. Sorry this woman is x squared this
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is one minus X squared minus X. To the
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fourth dx. This is the exact volume. However
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if you plug this into a calculator This is approximately
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.354 four. Mhm. Yeah. Always running.
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Sorry. 3.54459. Yes. Yes. Dude,
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I love it. Yeah. When they even approach
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a joke Yeah. Six. Which if we're rounding
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25 decimal places. This is 3.54460. Mhm.
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Of Oh no. Mhm. They're like I like
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like all right so that's part a in part B
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. Were asked to use the same curves but now
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we're going to reflect a rotation, say about the
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y axis. Yes that way. In the Greater
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Fort Wayne. Yeah. Now we're rotating a thin
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vertical strip about the Y axis. So we get
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a cylindrical shell. It says Veterans first response with
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the radius of X. Those flags and the height
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publishers which is you solve the circle square root of
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one minus X squared minus X squared Yeah. Try
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to carry. Therefore our volume is equal to using
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the shell method. The integral from 02 x equals
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our X intercept are right. Most X intercept This
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was the square root of route 5-1/2 mm.
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This is the same as from part A of two
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pi times our radius which is X times are height
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which is the square root of one minus X squared
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minus X squared dx kids. Yeah. For me
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. Mhm. Yeah, for America. Mhm.
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So this is the exact value of our volume,
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right? However you can use a calculator to find
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an approximate volume. If you do. The volume
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is approximately among trying to extract kind of branches weighing
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down his shoulders. He's just yeah. Yeah.
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Most Spy comes.9999. Mhm. Eight. Yeah
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. Two. Yeah. That would be from yeah
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. Yeah. Sure. Is there a guy who
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that's his? Yeah. Have a sturdy base your
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friend couldn't have. He would be your deal.
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