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Find the volume of the solid obtained by rotating about the x-axis the region enclosed by the curves $ y = \frac{9}{(x^2 + 9)} $ , $ y = 0 $ , $ x = 0 $ , and $ x = 3 $.

$V=\frac{3}{8} \pi^{2}+\frac{3}{4} \pi \approx 6.057$

Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 3

Trigonometric Substitution

Integration Techniques

Missouri State University

Campbell University

University of Michigan - Ann Arbor

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Let's find the volume of the solid obtained by rotating about the X axis the region and closed by the curves here in green. Why equals nine over X squared, plus nine and then y equals zero. That's the red line down here. Horizontal line X equals zero. This blue line over here on the left, an X equals three blue line on the rake. So here on the inside, this is a region that's being rotated. We rotate this thing about the X axis, and here's the reflection. And when we do a cross section on this thing, so let's slice through this volume vertically. We have a cross section that looks like this, and that's a Washington Excuse me, nothing disc and we could find the radius of this disc. So here's the sensor, and this is at some point X, and the radius is the distance from the center. So I equals zero all the way up into the green curve. So that that line right there, that's our So it's just this Y value, which is nine over X squared, plus nine because it's on the graft and then minus the Y value down here on the red line, which was zero. So this is our radius, So let's find about him. Pinnacle, they're ex clothes from zero three and then we have pie times, Radius squared and we're integrating Respect, Tex Solicitous at this point, looking at the denominator. X squared plus sign. Let's go ahead and take the transom. Three. Ten data, then the ex three seconds where, and we can go ahead and even simplify this expression. So, ex player, that's nine Tan Square. Let's pull that night outlets factory, and then it's also going to be square. So we have nine square tan squared, plus one using the protagonist identity for a tangent C can and then multiply that out so that would become the denominator. Also, we should go ahead and try to change those limits of integration in terms of data. Let's go to the next page. Sorry, trade sub. Write this down again. Recall that when you do a trick, self involved intention. The restriction for data is this one. And so let's go ahead and take the lower limit, which was X equals zero, plugging it into this equation. Here. I'm muscle for data, so this means tangent zero. And in this interval of here, the only time that happens is when theta equals zero. So that's our new lower limit, then. Our upper limit originally was three. Plug that in Frank's soft retention and the only solution here and this interval his power for So that's our new upper limit. So lower limit another room disquieted right out that interval. So taking out the eighty one in the pie now our limits are from the autopilot Before and then on top, we had DX after our troops up that became three sequence where there no detail and then the denominator. We simplified that on the previous page that was eighty one. See cancer the fourth. So here we can go ahead and cancel those eighty once in poetry and then divide, too, with the sea cans. So those go away that becomes a too. So we have three pie, see you in a pinafore, and then one oversee can square, which is co sign squared. So now we can go ahead and use half angle identity to rewrite co stands where? So let's pull out that too. Sierra Pirate Lore one plus Cosenza data, and we can integrate this. This is data Plus signed to that over, too, and our end Point's still zero floor. So for the last step, let's just go ahead and plug that in. So we have time before for Dana. Sign of pie over too, all divided by two. So this is from plugging in PIRA for And then when you plugging zero Data zero and sign of zero zero. So there's nothing to subtract. Sign a pie or two is one. So here we can go ahead and just combined. These fractions left three. Pi square over eight for the first fraction, and then when we multiply three pi over Sue till one half. We have three pie over four, and there's a final answer that should be afforded on there in the bottom of the one. That's it. There's an answer

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