solve-the-given-problems-by-integration-find-the-area-under-the-curve-yfrac1tan-1-2-x14-x2-from-x0-t

Question

Solve the given problems by integration.
Find the area under the curve $y=\frac{1+	an ^{-1} 2 x}{1+4 x^{2}}$ from $x=0$ to $x=2$.

Clarissa Noh · Accepted Answer

he is Claire&#x27;s. When you right here. So this is a tight turn into girl from zero to pi house for seeking of X minus tangent of ex Tietz. So we get the limit as a approaches pie hands from the left from there Oh too, eh, Seek. And the ex finest tangent of X T X to be equal to the limit of a approaches. Probably have from the left, bro. Oh, on second close tangent minus Elkton of sequined From their oath to a which is equal to limit as a approaches. Hi. I have from the left when we get Ln of seeking X plus tangent for her Second comes a route to a which is equal to the limit of the approaches I have from the left. Mia Ellen of one plus 10 engine over second from their oath to a get well enough to

Matthew Allcock · Answer

in this video, we&#x27;re gonna go through the answer. It&#x27;s a question of a 75 from chapter 8.7. So it has to find the area of the region that lies between the Kurds. Work was sec X Mike was connects from extra zero to X equals pi by two. First last drawer this area. So you go. Yeah. Quality. Yeah. Okay, So if this is gone, then Okay, let&#x27;s think about what sick that look like gonna start one run over. Think this the second X and then 10 X is gonna like Thetis. So we have been looking for the region. I&#x27;ve written what? The area is gonna be as an integral here. So given that we have this singularity that we&#x27;re asking toting towards, uh ah, X equals apart by two. Then we need to write this as the limits this be goes to pie like to from below off the integral from zero B, the second X minus 10 x the X. Okay, this is the limit. Oh, what we could use into the tables for the inter exact X. It&#x27;s gonna be a natural gas leak value. SEC X was 10 x similarly we find they had to do a motive for time. It&#x27;s gonna be minus, but not for long between zero and be. It&#x27;s not sequence of the limit of Write this as using standard, ah, rules of love with thems a lot with the miners with water with them and divide the arguments So sec X was time. Ex miner that divided by sex that&#x27;s gonna be one plus provided bicep zero that day, plus equal to the limit of the natural log. One plus 10. That&#x27;s it&#x27;s our necks because X then we divide them. That by SEC experts say, was times in by cause X that&#x27;s just getting it was sine X. It means A and B So we&#x27;ve now eliminated the singularity. So we could just write this as one plus fine. I see minus not your luck of one plus sign. Zero sign of 5.2 is just one so that we have the natural looking to Well, we have minus the natural log off. One side of zero is zero, but that&#x27;s a lot of one is a rose. This is just gonna be acquitted. Natural, too

Brittany Knowlton · Answer

So for this question were given that we have this curve, why is equal to a changing of X from axis equal to negative power for Teo X is equal to pi over three and we want to find the area between the curve and the X axis. So in order for an area under the curve, that is when we&#x27;ve integrate. So we&#x27;re going to the integral from our X values negative power for piratesthree of tangent of x t x. So here we are crossing the zero. So I&#x27;m going to separate this into two different in a girl&#x27;s native power for 20 of negative changing of x the X plus zero pirate three attention of X, The ex. So I want to integrate this In the book they did have what thes rules are equal to rich Hanjin. And it tells us that it&#x27;s equal tio natural log of seek it next. So now, plugging in my bounds, I get you natural log of seek it of zero Maya&#x27;s negative which is a positive natural log of seeking of negative power for plus plugging in the rounds. Here I get the natural August seek it, pirate three minus the natural log of seek it. Zero. So second of zero is just equal to one. So I get natural log of one here, plus second of negative pi over four is just a square root of two. Speaking of pie, over three isjust two minus again natural lock of one. That&#x27;s all I want is just equal to zero. So both those go away. Some just left with this here. This two&#x27;s really sir the one have powers to use my power. Roland want to go ahead and take out the exponents? So I&#x27;m just left with 1/2 a national out of two. Plus the natural order too. And now these air both like terms. I just add their coefficients. So I get three. House comes natural log I have to, and that is my area.

Bobby Barnes · Answer

we want to find the area between the curve. Why is even detention X and our ex access which is just wise? It was zero from exited with a negative pie for two X is equal to pi third. So I went ahead in Graf this region right here. And we&#x27;re interested in this area that I&#x27;m shading in red right here. No notice that we have where it flips. What function is on top of the other so we can call this region one. This region of here region too. So in the first region, Tanja is on bottom. Well, why is he zeroes on top? So let&#x27;s go ahead and write out what that is. That&#x27;s gonna be the integral from negative pie forth. And then this point of intersection right here. While we would just set this equal to zero. And we know Tangent of X is equal to zero at excessive reserve. So the upper bound zero and then it would be zero minus Tanja of ex, which is just gonna be negative tangent of X. Jax then So this is treated one and then for region too. So zero was on bottom and tangent of exes on top. So that would be 02 And they tell us to stop at pie third. So we just get tangent of X over on this side now. Well, we need to figure out how to integrate Tangent of X. So I&#x27;ll just do that off on the side is this We&#x27;re gonna do it twice, and I&#x27;m gonna first rewrite this a sign of X over CO sign of X. And what we&#x27;re gonna do now is to a use of where you is able to co sign of X and then we get the differential D&#x27;You zero to negative sign of x d x that we were to multiply this by negative one twice. We can use that to rewrite this to get the negative off integral of do you over you. Now we know that integrates to be the natural log of the absolute value of you, plus a constant C and then plug it back in you. I would give us coastline X plus c so we can go ahead and replace both of these here with that. Well, the 1st 1 would be the negative of that. So we should really get just the natural log. Oh, absolute value of co sign X. We evaluate from negative pie forth to zero and in the next one, then it will be the negative off go sign of X her natural log of coastline of X from zero to pie. Now, wouldn&#x27;t we plug in zero for this first interval? Well, co sign of zero is one natural log of 10 So the zero than minus so co sign of X or co sign of negative pipe, fourth coastlines and even functions. That same thing is co sign a pie Ford. So I should get this natural log of the square root of 22 Then we subtract off so co sign of pie. Third is 1/2. So that would be the natural log of 1/2. And we subtract off when we plug in zero, which should give us zero again. Now, let&#x27;s go ahead and simplify this down a little bit so we can go ahead and bring in that negative to reciprocate our natural log so that be natural. Log up to over route to which would simplify to natural log of just root, too. And then when we pulled this negative into here, Abby of us. Natural log of two. So let&#x27;s go ahead and move that 1/2 power outfront from the square roots that I give us 1/2 natural log on to, plus natural log of two. And that leaves us with three halfs. Natural longer, too, for the area that we have shaded there and rent.

J Hardin · Answer

here we&#x27;d like to find the area of the region bounded by the curves. Ten x tan squared X for X between zero power for So here&#x27;s a rough sketch of the graphs and let me explain why So, First of all, we know that can and tan squared or both zero on X zero and they&#x27;re both won when excess power for now in between for exes between zero pyro for we have numbers. So we have tan and tan square. So we have zero less than our people two ten eggs less than equal to one on the interval. It&#x27;s your own empire before, and I know that when you take a number between zero and one and you raise it to a higher power, their number itself gets smaller. So since the blue graph is a higher power of the same number ten, it&#x27;s going to be smaller than the red graph. And so to confirm this, let&#x27;s go to actual graphing calculator. So just this before the reddest hand, the bluest hand square and from zero to about power, for we see that the red graph is above the blue ref, so that tells us well First of all, we&#x27;LL have a formula for the area. So let&#x27;s write that out. This area is the integral A to be so here. Zero power before of the absolute value of ten eggs Linus Tan square. And by our previous observation, we noticed that tan squared was below or less than or equal to ten eggs. This tells us that tan eggs minus ten square Dex is positive. And we can use this because if so, if this is positive, that means that the absolute value is just itself. So we can write this as in a girl zero power before ten eggs minus ten squared. Now let&#x27;s go ahead and rewrite this zero power for CNX minus seek and squared X minus one. So we did here was use a path Agron identity to rewrite Tan Square a Sikh and squared minus one. We can evaluate all of these anti derivatives. The derivative of the tangent is natural log of absolute value of C can&#x27;t. Here we have a minus tan X, and that becomes so we have a double minus their sort of plus X and rn points zero power for So it&#x27;s going and plug in those end points. So is playing Pi over four first natural log seeking of Piper for his route, too. Since it&#x27;s positive we can drop the absolute value there and then tangent apart before is one plus x. So plus power for and then when we plug in zero, we have natural log and it&#x27;s seeking of zero is one tangent of zero zero and then plus zero. And we know that natural log of one zero So we could actually ignore this whole second term. And we&#x27;re left over with our final answer. Ellen Route too minus one plus pirate for So that&#x27;s our area, and that&#x27;s our final answer.

Problem

Solve the given problems by integration. Find the…

Problem 37 Easy Difficulty

Solve the given problems by integration.
Find the area under the curve $y=\frac{1+\tan ^{-1} 2 x}{1+4 x^{2}}$ from $x=0$ to $x=2$.

Answer

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Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus

Catherine R.

Anna Marie V.

Michael J.

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Precalculus Review - Intro

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

Catherine R.

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Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus