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Find the volume obtained by rotating the region bounded by the curves about the given axis.

$ y = \sin x $ , $ y = \cos x $ , $ 0 \le x \le \frac{\pi}{4}$ ; about $ y = 1 $

$$

V=\pi\left[2 \sqrt{2}-\frac{5}{2}\right]

$$

Integration Techniques

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Missouri State University

Campbell University

Harvey Mudd College

University of Nottingham

here we'd like to find the volume obtained by rotating the region bounded by the curves sign and co sign about the line Y equals one. So here's a rough sketch of the graph down here and read. This is the graph of sign from Zero power for and then in blue we have the Cho Sang graph. The green line is the axis of rotation. This is the horizontal line Y equals one And then the red and the blue graphs up top those air The reflections After we were rotated this area down here around the line Y equals one So we see after that we do is a revolution around the line y equals one we have ah, this hole in the middle of our solid. So that means that our cross sections which is indicated by this washer right here we'LL have holes So we have washers. Oh, so we have a formula for the volume in this case also, we obtained the volume by letting the washers move in the ex direction. So our follow ume will be in terms of X. So we know the volume is the integral. See you in a power before times the area the washer, which is pi times the larger radius squared minus the smaller radio square. So we need to find formulas for big R and little r. So it's relying the picture for this. So looking at the picture here, we want a large radius which is the distance from the center all the way out to the end point. So this is a big R. So one way to get there is to take this entire distance from the X axis two, the horizontal green line, which is one. And then if we subtract this distance right here from zero to the red graph, then informally, we have equality here. So we take the whole distance, which is one, and we subtract off the distance from X all the way up to the red. And we're left over with this big r. So we have big our peoples, the entire line which is one minus this. Why value here? But since this Y values on the red side graph, we could replace why with signings. So that gives us big r and then for a little R the same idea. We want the smaller line right here. This line segments a little radius, but we can get there by taking the entire line segments or the one and then subtracting the distance from zero all the way up into the blue graph. And then we're left over with lous little r So we have little r equals one minus y. But this same reasoning is before. But this time the y goes all the weapons for the blue graph. So this time why is co sign? So that's how we find big. Our little are in this case now we're ready to evaluate the integral. So first we should simplify as much as we can. It's quite implode that pie. So inside we have one minus to sign, plus sign square minus one. Then we have a minus minus to co sign and then minus coastlines where so we could simplify this a bit. So let's go ahead and we see that we could cancel the ones and we can apply Formulas should be a X there for science. Where and co sign squared. So use those double angle formulas with a pie in a girl's ear apart before minus two. Sign eggs, plus then this becomes one minus CO sex co sign of two X over too. We still have this to co sign and then we have a minus and this becomes one plus coastline to X over to. So we'LL need more room here. So it's goingto new new page so we can go ahead and cancel the one half in the minus one half in the previous page. So picking up where we left off, we have negative to sign X minus coastline to X over, too, to co sign X And then we have a new way says we have to times two of these. Let's go ahead and just put those together. Erase this. So this is just after simple simplifying the previous expression. Now we could integrate into these so we have to co sign X minus signed two eggs over too, plus two cynics and our end points or zero power before. So it's Quentin plug in these end points, so we plug in power for first. So co sign power for is rude to over too sign of pirates who is one? And then we have another two times room to over, too. When we plug in zero Ko san a zero was one. So we have two times one sign of zero zero. So these two terms go away and were we can simplify to get our final answer. Hi. We could combine these radicals to obtain too radical too. And we have minus a half minus two, which is negative. Five over two. So this is our volume, and that's your final answer.