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Numerade Educator

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Problem 14 Medium Difficulty

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 $

Answer

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

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Video Transcript

every in the show were given curves and a line and ratifying the volume of the solid obtained by rotating the region bounded by these curves. About this line were asked to sketch the region the solid and a typical disk or washer. The curves are Y equals sine of X. Y. Equals the cosine of X. X. Equals zero. And I guess also x equals pi over four. Although this is kind of redundant. And the line we're rotating is about the line Y equals negative one bertha sketch the graph of our region. Do we're having we're having gay sex bates, beautiful babies. It's beautiful babies, wow. Damn you. You could suck a dick. You can suck. We're some professionals. So first I'll sketch the curves. Y equals sine of X. This has a point at the origin and personal pi over four Is less than one. So it's it's about I guess I'd say here somewhere we increased like this. Mm. Yeah. Oh you know what? It's a mistake. So right We only increase to route to over two which is about here. Okay. And on the other hand, Co sign where we have the .01 and we decrease until we reach pyro. For route to over two also. So our region We also have the vertical lines X. equals zero. Yeah. So this region in red. This is our region. This is why I said x equals pi over four is a little bit redundant. My dad. Yeah. What gay people? Well, I guess not. We could have gone further. Now. The access of rotating around is this line Y equals negative one down here. Now in order to draw the solid, we're going to first reflect the region across this axis as most having gay sex in there. Yeah, I'll see you. Yes. All right. A real talk. So what? This is red dead fresh. Mm I've been playing Damn. You just killed a fucking bank tellers and my man just he ate that. Take that bullet like it wasn't shit. He said home. Right? Oh, so here is what the solid looks like, roughly. Yeah. Yeah. And if you look closely you can see that there is a washer which I'll draw in. Green harvest nuts. Who made this rock this star crossed? Yeah, missing something here though. That's better. I've been listening. Yeah. Mhm. Wait, it's no that's right. Oh the graphics and this just I love the graphic shit. You know. Whatever. Whatever fucking game journalists. Talk back to the David Connor frame rate. Yeah frame rate. It was back now. It's about she that's everybody's goal because it's not enough to just enjoy it. This is what the washer looks like. It's hard to see from this figure but our washer does appear to have an inner radius which is got a ticket the same thing. Well the sine of X minus negative one which is sine of X. Right plus one. Okay. And the outer radius patient is the co sign of X minus negative one which is co sign of X plus one. What? They only have the chinese in new Orleans and therefore using the watcher method are volume is given by pi times the integral from X equals zero to pi over four of our outer radius cosign X plus one squared minus our inner radius sine of X plus one squared. Now this is somewhat complicated integral so I'll walk through it with you. This is pi times the integral from 0 to Pi over four of Yeah this is the same as using difference of squares. Co sign of X minus the sine of X. Times the co sign of X plus the sine of X plus two D X. Mm. They understood that this is pi times the integral from zero to pi over four of co sine squared X plus to co sign of X. Some stuff, not to be too explicit here minus sine squared X. That's right -2 sine of x dx So and now using trig identities, I can combine to get pi times the integral from zero to pi over four of the co sign of two x Plus two times the cosine of x -2 times the sine of XDX. Now I'll take anti derivatives. So this is equal to pi times one half sine of two x Plus two Sine of X. Yeah, Plus to co sign of X from X equals zero to pi over four. Mhm. If you plug this in, we get the volume is equal to eventually Pi Times two times the square root of 2 -3 hats skipped a few steps here this year, result you all got pussy over.