Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.

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

$V=\pi\left[2 \sqrt{2}-\frac{3}{2}\right]$

Applications of Integration

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Numerade Educator

Harvey Mudd College

Idaho State University

Boston College

We're finding the volume off a solid attained by rotating that region bounded by y equal sign X Michael's coastline X I'm from, uh X equals zero two X equals four about the white donated one. So we can begin by drawing out the region we know from the very beginning that huh? The wind's gonna spend span from negative one to one, and that X is going to span from zero to high order for So we don't worry about the positive access. Call us one in negative one. Um, so you go first. Weikel Sine x Both of these curves relatively popular in common. So that is the sign. And the close anchor looks like this looks like the sine curve, but shifted exactly the same. But we only care about point where ex ranges from zero here. Two pi over four. So, actually, only they only care up to that part. So this is self installation. Here we're rotating about the y equals negative one mind. That's just this. And so what we get are washes. Now it's a little hard to see, but they'll curve that's creating The larger radius is the co sign curve and the curve that's creating the smaller radius is eyes the sine curves, so we have to keep that in mind. Now we're ready. We don't really care about the Y value at this point at the point of Interception, because we already know that the axe value is part before, and that's what we're integrating over from X equals zero t X equals, however, for so we don't go ahead and calculate the cross sectional area as a function of X, the big R, as I said before, it was created by the co sign Ex Kurt. So the distance between cosign X and the Y equals negative. One line is just the Y value of the curve. So the Wye Valley is cosign X plus one, because when you take into account this distance between the Y equals zero lion and the wife was new and that's always going to be one. And so the little R is just the signing skirt, and we're still adding one because there's still this distance that we need take care of. So about the pie in reverse, the order and really the only way to do this is the Multiply everything out So we get cause one squared axe It was too close An ex Why minus sine squared X minus two sine X minus one And so, if you know your trig identities, you'll recognize that co sign squared X minus side squared X is equal. You co sign of two ants so we can play. Don't make the calculations of the goal Simple. Keep the pie What we have. Before we still had a plus two coaster next and we still had plus one and minus one. Canceled. We started minus two minus two sine x. So we're gonna go ahead and put that into a nickel because that is R cross sectional area function. As I said before, we're integrating from X equals zero two X equals partner for well again. What about the pie from very beginning? Because it's a constant. There's two halves to co sign X. What is to sign X The ex? The value Anti derivatives. Now the anti drug order of coastline to X Well, the anti derivative of CO Sign X is just sign X. So we can just first begin by saying scientifics for cause me to ex. But then you have to remember the changeable. So the derivative of Sign to X is actually to co sign to X because you're multiplying by two at the very end. So, yeah, take care of that by adding one half at the very beginning. So that is the anti derivative of cosign to X anti drew of to cosign axis just to sign X and the anti druid of negative to sign. Next. There's just Teko sacks. All this is about the way Tio before and zero not before. Absolutely spy before you get pie times one half, not a sign sign of two times fire before two times before is party to sign of. However, too, it's just the Y value. So it's just the one credit plus two times no way to find a sign of fire before That's just three to two. Now it's evaluate co signed a fire before that is also true. Before it began, did you know that these two can cancel Eastern can cancel and that one doesn't really So now we can evaluate when X equals zero I'm distribute compiling this instance Nickelback, when X equals zero sign of zero is just zero. So we don't really care. Sand zeros, Just euro. We still don't care and co sign. Vex is one. So what do you care about that? I don't care about that one in the end. So what? We multiply this all out? Our final answer. It becomes pie. I want half plus to her, too. Because these two Americans together And then we'd still subtract two at the end. And so our final answer is pie. You're too, Linus. Three over too. That is our final answer.