WEBVTT
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a tourist is generated by rotating the circle X squared
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plus y minus capital are square is small, are
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square So that's this blue circle up here and conference
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one into we'LL rotate that about the X axis And
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down here we have another circle and the volume will
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weaken Take a cross section here and because of this
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large gap in the middle, we could see that
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the cross sections or washers And we we could obtain
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this volume by letting the washer trace out this volume
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in the ex direction where since the circle has really
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is our we have negative are over here the X
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value are over here. So these are our bounds
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of integration, Rex. And now for the washer
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We know that we'LL want usually the general formula for
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washers, pie and then large radius All due notice
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by are big square minus are small So this is
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it for using d X. So these Andy or
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the X pounds of integration Now we'LL have to go
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ahead and find big radius and small radius. So
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looking at the figure here we see that we could
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the radius Well, here's the center of the washer
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. Here's the origin in and to find the Big
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Radius it looks like we go up here, Toe
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are and then we have to add in that semi
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circle. So here's one way to find are big
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So let's look at this and my circle, but
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placed at the origin so that will be explored.
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Plus, y square equals R square And then we'LL
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go ahead and shift up the circle by our units
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. So after we do that, we shifted up
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by Are the circle looks like this. So in
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that case, we see that the total distance it's
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just are added to this upper half of the semi
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circle which is why equals from this equation right here
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from the circle R squared minus explode So are big
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We'll just be are the shift plus the upper half
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of the circle it's being radius and then similarly for
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the small radius are small Same idea but we'LL look
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at the lower half of the circle So this time
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we'Ll take that lower half circle and then we'LL shift
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it up I r and this will give us the
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smaller radius over here and dotage That's a small radius
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. So this time since we're on the negative circle
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our small B the shift up that we added in
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the vertical direction that's our And in this time it's
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plus. But we're doing the negative part of the
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circles of this time. It's negative R squared,
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minus explorers. So this is the equation for this
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Lower half someone circle exes. So this equation was
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the negative are slurred minus X squared. So we
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have found our big in our small, So we'LL
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go ahead and plug those into our volume formula.
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Let's go to the next page for that. We
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have v picking up where we left off and we
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observed that the bonds were negative. Arts are in
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the ex direction. We have pie. Let's pull
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the pie in the front of the girls. Before
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then, we have our Dade squared, so we
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have our plus the radical square the whole thing.
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So our big square and now we'll do our small
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squares. Then let's just go ahead and evaluate the
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squares so we have our square to our times the
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radical plus and then we square the radical. So
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this is the first square and then for the second
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square and then plus R squared minus x square.
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And then let's go ahead and cancel out as much
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as we can. So we have our square minus
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R squared. Those cancel r squared minus x square
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and then with a minus part squared minus X for
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years. And here the radicals to two are radicals
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Won't cancel because the double negative. So we get
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for our something, just go ahead and pull.
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The four are the constant for big Our pie.
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Yeah, in a rural negative. RTR and then
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we're left with Where was that? Excuse me,
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R squared minus little exclude yet I sees me.
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I pulled out. The four are here and I'm
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left with r squared minus x word. And now
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, with ready for a tricks up. So here
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you should take extra PR sign. What? Thandie
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Exes are co signed data. Now here we have
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to be a little careful. We are dealing with
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the definite general. So these bounds these limits of
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integration are subject to change. So to change these
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, let's go to our tricks up and recall anytime
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you sign entrance up years your restriction for data.
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So let's plug in these X values. So the
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lower bound X is negative arm and then the right
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hand side is our assigned data. This means signed
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data is negative one and in December hold on ly
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time that happens is when data's negative Piper too similarly
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for the upper bound X equals our plug. It
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in the left hand side becomes our and the right
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hand side. Our scientist, This means scientist is
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one and the only time that happens in our interval
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is that pirate too. So these air the fate
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of bounds of integration And let's go ahead and simplify
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this radical before we go on R squared minus x
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squared. This is equal to the radical R squared
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minus our square science where cool it are. A
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lot are square and then we have a radical still
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one minus sign square. This is our times a
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squirt of coastline square and that's just our co signed
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data so that the radical becomes our coastline data.
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But also DX gives you another Arcosanti data. So
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after multiply these arcos on Data's together so that'LL be
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our next date. Let's go to the next page
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we have for our pie negative, however, to
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two pirates who and then we'LL supplying the r co
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signed and our co sign We have r squared co
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sign squared data data. Now this little ours is
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not a variable That data is the variable so we
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can go ahead and pull the r squared out.
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And also for this coastline, let's use the half
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angle identity Coastline square is one plus coastline to data
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all over too. So pull out the two from
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the denominator So you have four over to which is
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too big our pie little r squared and then one
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plus co sign to data we can integrate this integral
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of one Just data in a rolls co signed today
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No scientific data over to and we have our bounds
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negative power, Hutu power Hutu And I will just
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put those in Data's pirate too. And then we'LL
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have sign of pie over Zoom And now we plug
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in negative however too plus sign of negative pie.
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Oliver too. We no sign of pie inside of
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minus fireball zero you have to capital are pie little
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r squared. Then you have pie or two plus
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a pie or two. So that's just another pie
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in there. So you'LL have it times pi And
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then we could simplify this by just squaring that by
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multiplying goes together and there's our final answer.