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