Problem

Volumes of solids Find the volume of the followin…

05:36

Problem 68 Hard Difficulty

Find the area of the following regions.
The region in the first quadrant bounded by the curves $$y=\frac{3 x^{2}+2 x+1}{x\left(x^{2}+x+1\right)}, y=\frac{2}{x},$$ and $x=2$

Answer

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Calculus 1 / AB

Calculus 2 / BC

Calculus Early Transcendentals

Chapter 8

Integration Techniques

Section 5

Partial Fractions

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Video Transcript

trying to find the area in Quadrant one bounded by thes two curves and X equals two. So I need to take a look at the graph of this and two over X Goes like that still in blue and the first function the three x squared plus two x plus one over the x Times X squared plus X plus one. That graph goes like this and X equals two happens to be over here. So the question is, where do those to meet? So we would have to set those equal to each other to find out the point where they intersect and then solve that equation, and we find out that they intersect at X equals one. Now that means we always want refining the area between two regions. We always do the top function in that Harry on that region, minus the bottom function. So we're going to be integrating between one and two of the three x Square plus two x plus one over the X Times the X squared plus X plus one minus the two over X and multiply that by the D. X. But in order to do that, I first have to take this initial function, and I need to decompose that I have to use our partial fraction decomposition and set that up. So we've got three x squared plus two x plus one over my denominator hopeful. Permit me not writing it. That's already factored. We have a linear factor of X, so there's going to be a hay over X, and we have a single quadratic factor, the X squared plus X plus one that could have a X term and a constant term in the numerator. So we have B X plus C over the X squared plus X plus one. Now we multiply all that by the common denominator to get rid of our fractions. And the left hand side just gives us the three x squared plus two x plus one on the right hand side. Each of the factors in the Dominator will divide out, leaving us with the numerator times the opposing denominator. So it's eight times the X squared plus X plus one gives us a X squared policy exploits. A the B x plus c Times X gives us be X squared plus C x, and now we can equate the terms and the coefficients to figure out a B and C so doing the square's first. Yeah, three equals a plus. B doing the ex terms we have two equals a plus e and then wither. Constant term would have been nice if I would have started there. We have one equals a and if one equals a, then that means C equals one. And that means that b equals to. So now I can write my integral again with the decomposition and the subtraction. So we're integrating from 1 to 2, uh, a over X one over x plus b x plus e two x plus one over the X squared plus X plus one and then minus the bottom function. Two over X and that's all Times X. I can combine my one over X minus two over X. So if you will permit me to just the race and that will become minus one of Rex. So now to integrate thes integrate each one individually, integrating the first function with R D. X on there. Notice that if I let u equal the denominator x squared plus x plus one do you is gonna be two x plus one. So that means when I put my TX on here with my two x plus one And I have two x plus one d x here This is in the form of one over you times do you? So this is just gonna be our natural log in a girl. So we will have the natural law off the absolute value of X squared plus X plus one and then the one over X DX. That's obviously one over Utd you So that will be minus the natural Log off. That's what value of X and that's gonna I'm gonna combine that into one logger of them. So I will have the algorithm of X squared plus X plus one, all divided by X and absolute value. And now, to find our area, we evaluate that between two and one. So put those in and we have the while. Rhythm off two squared plus two plus one divided by two is seven hands minus of algorithm off one squared plus four impulse warm, divided by one is a lover than the three again I can write. This is division over the mob. Seven have divided by three, which ends up being the algorithm off 76 So that would be the area

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