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Find the area of the region inside all the leaves of the roser = 3 sin 20....

Question

Find the area of the region inside all the leaves of the roser = 3 sin 20.

Find the area of the region inside all the leaves of the roser = 3 sin 20.



Answers

Find the area of the region.
$$y=\sin 2 x$$

Okay, So for this problem were asked to find the area of all three leaves of r equals to co sign of three data. So I have to graph here, but I want us to look to see if we can determine if the interval is pie or two pi. So I'm gonna plug in these values, so co sign it. Zero is one. So two times one is two. And so at two with data is here on the graphics on the dot on drawing and then three times pi over six becomes pi over two. So, co sign of pi over two is gonna be zero. And so on the graph, you're going down 20 And this is where pi over six is on the graph. Okay, two times pi over 333 times pi over three is pie. We're co sign is negative. So it's gonna be negative, too. And so on the graph, if we were to draw this out, this is where pi over three is. So the negative value of that is going to be here three times. Pi over two is gonna be three pi over two. So co sign of three Pi over two is zero. And as we can see on the graph, we're going back to zero three times two pi over three is two pi co sign of two pi is one so it's gonna be too. So if we're going to two pi over three, which is going to be right here, we're going upto one and then five pi over six times three is gonna be five pi over two. Which coastline is going to be zero. We're going back down to zero and then co sign of three pie is going to be negative one. So then radius is gonna be negative too. And since pie is here, negative to here would be back to the beginning. So this tells us that the interval for this particular function on the graph one full rotation is from zero to pi. So what I'm going to do now is I'm going to you look at the formula, so it's gonna be from a to B. R squared are sort of 1/2 R squared detail. So for this, we're gonna be using the interval from pie 20 or zero the pipe and then 1/2 and then we're gonna be using to co sign of three data squared de fainter. So I'm going to go ahead and square this. So I have my 1/2 So two squared is four and then coast sine squared of three data. Now coastlines squared, I cannot use. So I'm gonna use the the half angle identity, which tells us that it's 1/2 1 plus co sign of tooth data. Okay, so 1/2 times four is gonna be, too. And then for my co sign identity, I'm gonna use this. So 1/2 and then one plus co sign instead of doing to think it just means I'm doubling what the data already is. So since this is three, it's gonna double to become six. So two times 1/2 cancels. So now I'm just left with one plus coastline of six data, some of the integrated. So the integral of one in terms of that is just gonna be fade up. The integral of co sign is going to be sign of six data member coast. Integral coastline is just signed. And then I have the six in front, so I've gotta multiply this by 1/6. So if I were to derive 16 sign of six Data is gonna become co sign of six data. So then data says will become pie, and then sign of pie is zero. Then, of course, zero sign of zero is zero. So my answer is going to be pie.

According to the question, we need to find out the area we have given The question is are is it will do sign for $2 when the soul we know that the form love area is one by two integration, it will be our square dpt Then we used these value we have is he could do one by two limit is zero toe by We know that when Article 20 sign for two days equals zero and sign $40 is equal to zero. There is limits zero and by in case off sign 00 signed by zero. We know that there is our square. It means Science Square 432 Did you tell? We know that sine squared Trudeau is equal to one minus course booted up. Ah, born to when we applied this property in this Tom, we have is equal to one by 20 to buy one minus course Air Ceuta upon to digital when the soul we have is equal to two to Dr Miners one up on it. Sign it could, uh, zero toe by by four. This is a 14 time. There is area off to buy a bone started to when the soul we have by phone 16. We know that though value off area is five by 16.

Okay we are going to find the area of the shaded region. So this guy right here that's created. The shaded region that they want us to find is actually this bottom pedal. But it is created between um pi over three and two pi over three it just happens to be negative and so it's across to the other side so we can do our one half of that range of values. And then we're squaring are are so we get a four sine squared of three. Theta. D. Theta. Okay so we're going to have to use a power reducing formula. We can't integrate sine squared but we can definitely integrate um a co sign of six data. And this power reducing formula does double the angle. And then we also have some other things that we can clean up here we have the one half and we have the force I'll bring it to out front and then I'll bring in my new formula which is a one half minus one half. Co sign double the angle. So we're at six data. So we're ready to integrate. We can also consider when we multiply those twos in um that both those halves will just become one. So our one constant when we integrated in terms of data becomes data and then we have a -1 co sign of six data. Well co sign is going to go to sign but where extra 16 will come in through use substitution. So now we can put in our values that we can put in to pi over three. Now when we do put that into the sign of six pie, that becomes um for pie and sign. There is zero. The same when we put the pi over three in and then we say minus, but that also becomes zero. So it's really just to pi over three minus pi over three. So our answer for that one pedal is pi over three.

Okay, So for this problem were asked to find the area of the of each. The entire function of r squared equals six. Sign of tooth ate up. So for this one, what we need to take a look at as we have two leaves and by doing a little bit of an investigation, we could plug in the values for zero and find out that one of the leaves goes from zero two pi over two before hit zero again. So what we want to do is go ahead and do find the area of the integral from between A and B of 1/2 R squared in terms of NATO. So since we already have r squared and we now know that it's between pi over two and zero, then we can do 1/2 now. Also, one other thing before we going is, since we're this is only for one of the leaves. I wanna multiply that by two, and then r squared is gonna be just six sign of tooth data. So since my two in my 1/2 cancel each other, I'm just gonna integrate directly. So sign it's gonna be negative. Co sign of tooth ate up, and then this. Then I'm gonna have the six in front of it. And then I'm also gonna multiply by 1/2. So one halftime six is going to be three, and then I'm going to apply my values of pi over two and zero now, co sign of two times pi over two is just simply pie. And so, co sign of pie is gonna be negative one. So we have negative three times. Negative one minus co. Sign of zero is one. So we have negative three times one. So this is gonna be three minus negative. Three. She's going to plus three. And so the answer is going to be six for.


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