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

$ y = \sec x $ , $ y = \cos x $ , $ 0 \le x \le \frac{\pi}{3}$ ; about $ y = -1 $

$$

\pi\left[-\frac{\pi}{6}-\frac{\sqrt{3}}{8}+2 \ln (2+\sqrt{3})\right] \approx 5.95 \text { units }^{3}

$$

Integration Techniques

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here. We like to find the volume obtained by rotating the region bounded by the curves C can co sign for X between zero and pi over three around the line y equals negative one. So over here on the right, there's a rough sketch of the graphs and read up here. In the first question, he was co sign from excess from zero power three. So coastline starts off at one and then co signed off. However, three's a half, and here in blue, we have seeking. Speaking of zero was one and then c can of pirate three is too. Then we have the line y equals negative one, and we reflect this region between the curves, the blue and the red around the line y equals negative one. So here's a recollection down here in the fourth quarter, so red is the reflection of the coastline. Blue is the reflection of the sea can, and here we see, we have a cross section, which is a washer. So this tells us that the volume will be of the form pie times of the large radius outer radius of the washer, minus the owner or the smaller radius the washing and the reason will use the X is because the volume is obtained by letting the washer moving the ex direction from zero to five for three. So next we have to find expressions for capital R, and little are so let's start off with upper case are so coming over here is our picture We see ours the outer radius. So here's the center of the washer and red right here and then we obtain the outer radius by going from the centre all the way up to the outer edge of the washer. So this is our upper case are and we see that this lower portion, this first part of our down here I'm scribbling in blue. That's just the distance from negative one zero. So that starts off. There's just one. And then this upper region up here, it's just a Y value because it's just from zero to a graph percents. This why value is on the blue graph it seek and so we could replace why with seek Innovex and similarly we could find a little R. So let's label little are on the on the washer, so we start off from the center of the washer. And we go until the first curve that we hit, which is right there and read. So this is little R. So this first part of that are down here in the bottom below the access excesses scribbled in blue, that's just one. And then we go from zero to this y value on the group. But this time we're on the red curve, which is close on so that that those are expressions for the the radio guy so we could go ahead and plug goes in and then find the So we have You couldn't pull that pie. Okay, that was good. And evaluate these squares Well, pie in a girl sort of laboratory. Then we have one plus two thinking. Plus, he can square minus one minus two co sign minus co sign squared, see if we could cancel in here. We see those ones, cancels, Let's cross those off. Also, we can rewrite this co science. Where is one plus two co sign of two eggs over too s. So this is the double angle formula for co sign, and it'LL make it easier to generate the other terms or we already know the intervals of sea can of seek and squared and co sign but co sign squared will be easier to integrate after replacing it with this term over here. So let's go to a new page. We need more room. So picking up where we left off Pi integral Part three. Then we have sequence where plus to see again. Minus one over two minus coastline to x over too minus two co signed X So let's go ahead and integrate each of these. We know the integral of sequence where this tangent in a girl of Sikkim is. So here we have the two and then we have natural log Absolute value. C can't plus tangent. Then for one half becomes excellent too. And for the coastline to X, we have signed to X over two times two, which is for so again here. If this two exes throwing you off, you could go ahead and use a U substitution. New equals two X And then finally this negative to co sign becomes a negative to sign X and the end points are zero pirate Drea. So it's good and plug in these points. So tangent of Pirate three's Route three so grew three plus two times natural log, absolute value. Seeking a power three is too. So we have two plus radical theory and then minus pi over three, divided by two since Piper six and then we have a minus sign of two pirate three. So that room three over too, divided by four. So that would be rude. Three over eight and then minus two times, sign a pie over three fifty times. Room three over too. So this is after plugging in Pirate three. And then when we plug in zero, we have tangent zero zero with two natural log seeking a zero was one tangent zero zero minus zero over too. And then sign of zero zero. So we have minus zero minus zero and we know that natural other born zero. So all these terms are zero. So now we just simplify the expression that we have also notice Over here these will cancel. So it's combined these radical theories. So doing so, Combining those fractions we should get negative Route three over eight. We still have this minus five or six and then plus to natural log two plus route three and we could drop the absolute value here because two plus room three is positive. So no need for absolute values here. So this is our final answer, and that's the volume.