solve-the-given-problems-by-integration-find-the-area-under-y6-sin-2-x-cos-x-from-x0-to-xpi-2-3

Question

Solve the given problems by integration.
Find the area under $y=6 \sin ^{2} x \cos x$ from $x=0$ to $x=\pi / 2$.

Audrey Fong · Accepted Answer

in this question we want to find the area bounded by this curve And x equals to zero to tie over to In general. When we want to find area under a curve with X. From A to B. We will just take a small strip over here. And the wife of this trip is so small it&#x27;s just a small change in X. Or quite D. X. So over here it has a wide value. Why? So Ariel. This small strip is why the X. Yeah. So to find a total area under the curve right here we&#x27;re access from A to B. We will just sum up. But because this is a continuous submission it will be an integration sign something from A to B. All the small strips we have so this will become area underground. So back to this expression. The area will be from Mexico 0 to power to Y. G. X. But the one is 69 square X. Course in X the X. Now there are a couple of methods to integrate this. But for this topic we&#x27;re using substitution method and the integration power rule for the power rule. If I integrate you to the power and respect to you I would want to the power and divide by the new power for indefinite integral. We need to put plus C. For definite integral. We do not need to do so. So let&#x27;s use substitution. We can next you be equal to sine X. Since there are more sign then co sign here the U. D. X will be called X. So the U. Is co sign X. D. X. So this part is the you. Now we need to substitute the lower and upper limit because these are in X. We want to make them in you because little be using you. So when X equals to zero the lower limit My you is signed zero and they&#x27;ll be zero. Yeah When X equals two pi over through the upper limit. My you will be signed pie or two. And there is one So we&#x27;re ready now to substitute. So the zero X equals zero. Now becomes you goes to zero as it goes to power do now becomes you go to one. Now these six can be brought out. Science square X is now since I X is used or sine square eggs will be used square Jose asd excess. Now, d&#x27;you Using power rule we will be integrating new square. So using the power rule you square. The power will be added by once every three divided by the new power. Yes. This and this cancer leaving us with two. So we have to you can also stopped the upper limit in and then minus off by something. The lower limit in. So be one cute minus zero. Cute. So the final answer is to you need square

Ian Maurer · Answer

Okay, so here we about are integral from pie over 62 pi over too. I mean, we have signed up to X divided by two scientists. Now, what&#x27;s going on here is we have two different kind of types of sign, so I would just look up your table, the trig identities. I&#x27;m looking for something involving sign of two acts. So that&#x27;s our double angle sign of two X is to sign ex CO sign X. I&#x27;m just gonna This is what I saw in my table of identities. Sign of two acts. ISS too. Sign ex co sacks, so I&#x27;m gonna substitute that for the top of the integral pi over six. High over to excuse me too. Sign ex co sign. Xs are top bottom. I haven&#x27;t done anything with so to sign X, the X and that&#x27;s really nice to use. Cancel out science camp full out. We really just have to do the integral from pi over six by over two of CO sign X, the ex anti derivative of co sign this sign. So you have sign of X from pi over sticks by over two clubbing in the top value first sign if I over to plug in the bottom value. Second sign a pie over six and evaluated. Sign of a pirate or two is one sign of pi over six is 1/2. Final answer is not negative. Sorry. Just regular 1/2.

Vipender Yadav · Answer

here in this problem, we have to evaluate the integral zero by by six Underwood off one place. Co sign two weeks not be X. We known that, uh, consign Twix. Is it? Calls to co sign Square X plus one? Uh, actually, it should be minus one, so the end result will become ill. Use this for my hair on. It&#x27;ll become zero to buy by 600 off one place Lucas in square X, minus one dot dx one and one candle out. So I&#x27;m thinking the route Goats aid on a new road off Assigns to Alexis Kasai Knicks not MDX on the LTD&#x27;s from zero to pi basics on the integration, of course, I nixes signings on the limit is from zero to, uh it is by basics on route who&#x27;s outside so simply go to problem it. So roach, two times signed by by six is one divide by two on ST zero is zero. Finally, we can conclude that in Tegel zero to buy by six under Rudolf one place co sign two weeks not DX is equals to route to divide by two. This is the final answer

Enkhzaya Enkhtaivan · Answer

So in this problem, we have internal from zero to, um internal from zero to pine for six. Skirt of one class sign. Thanks. Well, so this interim squared, remember, sentences is a bit tricky, but eh So basically, the intuition is simple, though whenever you see a square, the function you want to get rid of two square, so it means you want to right one per sign off X as they square something. So some nice work in function. And so there&#x27;s only a handful of formula we know about side effects. Which double angle formula are the half in a form us really? And if you play around with him, you&#x27;ll see that the half angle formula is going to be useful, right? And after this, well, you have this constant one here, and whenever you&#x27;re doing with trick functions and you see the number one, the first thing you should be that should be coming to your to your right is the famous sign squared plus co sign squared formula. So now we have this plus to co sign X over too Senate X over too. And then you would realize that this is the square of course, scientific, sir. Two signers, except for two. Therefore, our in through Integral immediately becomes something rather simple. So we were We retake the squirt. It&#x27;s It&#x27;s just simply Chris sign D x here, and we can do the antidote. There&#x27;s immediately. So this will be sign of X over, too, while two times as two times cursed sign off X over to and the valued at points zero six. So let&#x27;s do this real quick. Um, Howard has two times sign off. Hi. Over 12. Thank you. Two times co saying pi over 12 and that the lower and the 0.0 which will want to give Oh, negative. Minus minus two. It comes from the co sign of zero and the sign of Serious Just here. Now, technically, this is a miracle answer. So we can&#x27;t leave it here. But as it turns out, this proud and we couldn&#x27;t even simplify this furthers using some clever trick formula. So it&#x27;s trying to do that. So I&#x27;m going during the next speech here. So now this is basically equal to two times. No, I have it went to now, too. Course, And I have this whole thing, so That&#x27;s right. There&#x27;s two tires squared Subasic and want to plan by squirt of two and square toe the line of scrimmage to so they don&#x27;t have whenever screwed. Too sign of higher, too. And then one divide script, too. Cursed sign. All right, 12. Well, you could you could do this without this portion, Annie. And if you knew the numerical clothes from River Seine and personify were 12 What? That&#x27;s something most people haven&#x27;t memory. So we&#x27;re trying to figure out it this way. And now the kid kid trick is that to recognize them, this one over square of two is actually sign off. Pyro. For similarly, this is co sign off piles before. Now you see where I&#x27;m going with its right. So this will become two plus two times square of two times. Um so it&#x27;s one more time. Sign of high over for sign off. Pie over 12. My Askar, Sign off pie off the floor for a sign. Hi. 12. And this whole thing is actually equal to negative co Sign off. Hi. Over three. Why? By the sum of anguish Formula. So you, Khun Khun checked. So this will become to the whole answer becomes now to purse two squared to And of course I have pie or three. It&#x27;s Christian of 60 degree, which is just one house. So with the negative one house, this is equal to that. So finally, the answer is to bring a squirt of two.

Ernest Castorena · Answer

Okay. It&#x27;s time to give us the Internet into Rome from 0 to 2 pi over two and then from white. It&#x27;s a pi over two of the surface. Six times the sign two X minus three Way the ex do you want? Well, let&#x27;s see. Let me verify. That&#x27;s correct before anything else. Okay? Written correctly. Now they want us to sketch the region of integration. So let&#x27;s do that. Are wise from zero Teoh high over to. So we&#x27;re gonna really getting in a girl from my toe. Love, Rachel, we have a vertical line. X is equal toe high over tea and then we just have a lion ways Equal tanks. So they want us to compete this region. Let&#x27;s do that. Zero. Hi. So this time I suspect eggs, so we&#x27;ll end up with negative. Let&#x27;s see, six over negative three co sign of two x ministry way Evaluated why? And high over to. And let me just verifying that you take the dirt of one that with negative same first negative cancels out and then we won&#x27;t supplied by a negative three. So the other negative three on the bottom gets comes about only ended with 16. Okay, it looks so now let&#x27;s evaluate. So get are single and go single veil by variable. Wonderful. See the negative science cancel out. So we have a to co sign? Um, let&#x27;s see. High minus three away and then weariness of trying to no sign of this time plug in white. We end up with a negative y the way. But Carson is even so, we can just leave that as a bus. And now let&#x27;s take our so the ant eater to sign of pi minus three way. But we&#x27;ll need to divide by a negative. They&#x27;re here. I&#x27;m in Over here. It will have a minus. Two sign of why Evaluated at zero on by over two. Okay, lets see. Well, if we value that zeroes in terms definitely zero. And the first term of sign of pie, which is also zero. So the only thing we&#x27;re concerned with is the pi over two. Let&#x27;s see what we could get. Negative 2/3 times. Sign of Let&#x27;s see pi subtract three pine over to Well, I just turns into a negative crying over two, and then we can pull the negative out because it&#x27;s odd It&#x27;s a lot on council. A sign of pi over two is just one. We just got a one here on our second term. Will have, Let&#x27;s see minus two times ones inside of by. Over two. It&#x27;s one. So end up with two there minus 6/3 when we get negative for over three.

Amit Srivastava · Answer

and give it. Could I? 56 252 And, uh, it is got six long sign. It&#x27;s soup course it&#x27;s log the site. It&#x27;s yeah, if we consider Let&#x27;s sign X equal to P. So wow, cause sex here because it&#x27;s cause the X now change the limit. They feel put X equal to five or six. It will be have X equal to vibrato. They will be one. So lower limit is have upper limit is one causes yesterday Opportunity. So log. Okay, duty now using buy parts. And I have one log T and one as a second something. So finally, direction up by part will be lucky thinking minus f incision off Lucky 50 Dickie and our Saturday insert It will be love tea where it is one Burti beauty. Well, beauty and this cancer. So this is the love p minus and porter limits one. And how so? One valuable Put one Love one with zero minus one and minus have lord one very toe minus. So by this one, unless one minus one by two. Don&#x27;t secure Dickinson to love too

Problem

Solve the given problems by integration. Find the…

Problem 34 Easy Difficulty

Solve the given problems by integration.
Find the area under $y=6 \sin ^{2} x \cos x$ from $x=0$ to $x=\pi / 2$.

Answer

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Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus

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Allyn J. Washington, Richard S. EvansBasic Technical Mathematics with Calculus

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Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus