find-the-area-of-the-shaded-region-by-a-integrating-with-respect-to-x-and-b-integrating-with-respe-2

Question

Find the area of the shaded region by (a) integrating with
respect to x and (b) integrating with respect to y.
(Graph cant copy)

David Marsella · Accepted Answer

So here we have. Why equals X squared minus two and y equals X, And we want to find the area of the shaded region. But we&#x27;ve been asked to just set it up. We&#x27;re just going to set up the integral we&#x27;re gonna set it up for both. With respect to X and with respect alive, we don&#x27;t actually have to solve the integral, which is gonna save us a lot of time, actually. So first, let&#x27;s look at this. We know that our boundaries are the X axis, the line y equals X in our parabola X squared minus two. And we know we&#x27;re gonna need these three points to help set our boundaries for any girls. So let&#x27;s go about finding those points first. We know this point is 00 because y equals X passes through the origin. That was easy enough. We&#x27;re gonna need this interception point where y equals X squared minus two passes the x axis. So if we set y equals zero there we get zero equals X squared minus two, which factors? Two X plus radical too and X minus radical, too. So we have the excess intercept at X equals plus or minus groups, plus or minus radical, too. Now, our point is in the negatives we&#x27;re not We&#x27;re not worried about this point here in the positives. So we care about our negative. So the point we&#x27;re gonna use is appointed negative, too. So this point here, his negative radical, too zero. And then we need this point here, which is our intercept point. So intercept is where they&#x27;re equal. So we get X equals X squared minus two or zero equals X squared minus X minus two. And this factors two X minus two and X plus one. So we have intercepts at X equals two and X equals negative one. Now, again, we care about our negative x value because that&#x27;s what Quadrant Warren. So we&#x27;re not concerned with our positive value were concerned with our negative value. So we care X, isn&#x27;t it minus one? If we plug X is minus one into y equals X, we get negative one and one. So that&#x27;s our final point here. Negative one negative one. So we have our three points where you&#x27;re gonna represent our boundaries, and we have her functions. So let&#x27;s look first with respect to X We&#x27;re gonna set up an integral we have A equals. The area equals three integral from A to B of F of X, minus G of X. And if we consider, are vertical line test here. We&#x27;re gonna need to different inter girls and they&#x27;re gonna be split here at negative one. So let&#x27;s set those up. We have the integral from negative radical, too to negative one. And our top function is just the X axis. So it would be zero and are g of X are bottom function is gonna be y equals X squared minus two. So he would actually have zero minus X squared, minus two. So we&#x27;re just gonna put a negative sign out front. We have negative X squared minus two DX plus our second integral, which is gonna go from negative one 20 And here, where y equals X is our G of X. And again, our f of X is simply zero. So are integral is going to be negative. X dx. We can simplify that a little bit. There&#x27;s very little change, but we&#x27;ll do it just for your simplicity&#x27;s sake. It&#x27;s the integral negative radical to negative one of two minus X squared DX minus the integral from negative 120 of X DX. Both are correct. Both are set up on. That&#x27;s all we need to do. Now. If we were to do this with respect to why our area is integral for May to be of effort, why minus g y de y, uh, And if we look so what we&#x27;re gonna need to do here is we&#x27;re gonna need Teoh get our equations with respect. Why so y equals X simply becomes X equals y and y equals X squared minus two is going to become X equals plus or minus radical y plus two. Now again, we are in the negative quadrants where X is negative and in fact, the region wearing his exes and wire negative. So in this case, we care about negative radical y plus two because this is going to represent this half of the parabola that&#x27;s on the left side of the the Y axis. So that&#x27;s what we&#x27;re gonna use. We have X equals Y and X equals negative radical Y plus two. Now, if we look here and we do the horizontal line test for with respect to y we noticed we only need one Integral because we&#x27;re going to hit the same two functions all across our boundary. In this case, our boundaries gonna go from negative one 20 So are integral is from negative. 120 r f of x is our right most equation, which is why equals X. It&#x27;s our line. Or, in this case, X equals y Why minus negative? Radical? Why plus two. Do you, I or Teoh simplify the negative signs? We have the integral from negative 120 of why plus radical y plus two de y and that&#x27;s it. We don&#x27;t actually need to solve these into girls, but we&#x27;ve set them up.

David Marsella · Answer

So here we want to represent the area of the are shaded region with two inter girls, one that shows the area with respect to X and one that shows it was respect to lie. So we&#x27;re gonna need a number of things before we do that. To start we&#x27;re gonna need are three intersection points. So let&#x27;s begin. We&#x27;ll start with our parabola where it crosses the X axis. So zero equals X squared minus four X or X Times X minus four. So we have our X intercepts when X equals zero and four. So that&#x27;s thes points here. 00 and 40 and then we want our intercept points of the line in parabola. So X squared minus four X equals two x minus eight He comes X squared minus six X plus eight equals zero, which factors into X minus four and X minus two. So the two functions intercept when X equals two and four. We already have the point. When X equals four R point down here becomes to put two into an equation. We get four minus a negative four. So we have our intercepts. A quick look at our parabola, our peak of our parabola is that too? So this line intersects our parabola right at the peak. So it actually splits are parable a perfectly in half at the point to negative four. Uh, this is relevant in a minute because now we need to get our equations in terms of why? Because the second half of this problem is going to be expressing her integral. In terms of why so are y equals two x minus eight. It&#x27;s fairly straightforward. We end up with why plus eight over to equals X for our parabola, we have y equals X squared minus four x. We have to complete the square. So yet why plus four equals X squared minus four x plus four. This the ex polynomial factors into X minus two quantity squared. We take the square root of both sides. We have plus or minus radical y plus four equals X minus two or X equals two plus or minus radical. Why plus four Now the reason It was important that this pointed to four perfectly by sex. Our parabola is because now we can look at our two plus or minus radical y plus four and know that we only care about 1/2 of this problem. We care about the negative half. So the equation that is relevant to us is gonna be X equals two minus radical Y plus four. That&#x27;s the one we&#x27;re gonna need. So let&#x27;s let&#x27;s actually set up our inner girls first will do with respect to X, which we know the area with respect to X is the integral from A to B of f of X minus G of x dx. If we look at our area, we&#x27;re gonna have to regions split here where X equals two. Because with our vertical line test, we have two different functions that we&#x27;re gonna hit on our lower function. So we&#x27;re gonna have to different g of X. We&#x27;re gonna need to different girls. So our first area is going to be from 0 to 2 and our f of X for both of these is going to be Why equals zero. So our f of x zero. So we have zero minus X squared minus four x DX, plus our second integral, which is going to go from 2 to 4 zero minus our line, which is two x minus eight DX and that&#x27;s it. We&#x27;ve got it set up. We can simplify it a little bit. We have the integral from 0 to 2 of four X minus X squared DX plus the integral from 2 to 4 of eight minus two x dx. And there we go. That&#x27;s our area. With respect to X, now we can move on to our respect, toe Why? And we know here that are area is the integral from a to B of f of why minus g of why do I? And if we use our horizontal line test, we can fill this entire region by hitting just the same f of X and G of X, so only require one integral. So it&#x27;s gonna be from our negative four. Because remember, we&#x27;re in respect to why this time 20 our right motive, right? Most equation is going to be f of X. And that&#x27;s our line. Which is why plus eight over to minus our left most equation, which is our two minus radical. Why? Plus four. And that&#x27;s in parentheses. Do I? We&#x27;ve set up both inter girls. Oops. So here it isn&#x27;t respect. Why? And here it is a respect X

David Marsella · Answer

So here we have a parabola X equals y squared minus three and a line X equals two y in a region that we want to express only in terms of the inter grills. And I want to do it both with respect, X and respect Why we don&#x27;t actually have to solve the inter girls. So all we need to do is set up. So first off, we know we&#x27;re going to be doing with respect to X and y we&#x27;re given our equations with respect. Why? So let&#x27;s just quick get them in respect to X as well Our exit polls to why is gonna become y equals X over two and her X equals y squared minus three is gonna become why equals plus or minus radical X plus three. So we have them in terms of X and Y, and we know that is part of this region we&#x27;re gonna need these three points has boundaries for our inter girl. So we need to find those three points next. Uh, the end of the parable over here is just going to be whenever y equals zero. We said, why equals zero? We get X equals minus three. So this point here is negative three zero and then the other two points or the intersection points of the parable in the line. So we&#x27;re gonna set those equal to each other two y equals y squared minus three. We can reorganize that into y squared minus two y minus three, which factors into why minus three times y plus one. So these intersect when y equals three and negative one. If we played those into, uh, one of the equations, we end up with the points 63 and negative to negative one. So this is the points that&#x27;s gonna that are going to represent our the boundaries for integral. We have all over her equations. So now we can set everything up. First, we&#x27;ll do it with respect to X. So we know the area is the integral from a to B of f of x minus G of x t X. We know we&#x27;re gonna have to regions, because if we do a vertical line test, we&#x27;re gonna end up with a switch here, where on this side, we&#x27;re gonna have two different functions. And then on this side, the proble are vertical line or or are vertical line is going to hit the parable on both sides. So we&#x27;re gonna have to supper regions. Our first region is gonna be from negative three too. Negative too. And actually, I&#x27;m going to erase this real quick cause I&#x27;m gonna need space. One thing we know this parable a is symmetrical about the x axis. Uh, if we do f of X minus g of X, we&#x27;re gonna end up with radical X Plus three minus negative radical X Plus three, which is just radical X plus three plus radical X Plus three. Or it&#x27;s just two times radical X Plus three. Another way we can think about it. We know it&#x27;s symmetrical. We can just cut it in half weaken, do the positive side and then you multiply it by two. So the area is two times the integral from negative three Too negative, too, of radical X plus three de eggs, plus our second region which is going to go from negative to 26 And our f of X is gonna be the top half of our parabola, which is positive. Expose three minus R g of X, which is our line which is X over two Dix, and that&#x27;s it. You&#x27;re done with that part. We have expressed it in terms of an integral. We don&#x27;t even need to solve it. Next. We&#x27;ll do respect. Why we know that this is a similar integral that goes from a to B and it&#x27;s f of why minus g of lie in this case, our horizontal line test, where along the whole area of this region, we&#x27;re only gonna have one region. It&#x27;s going to go from negative one, 23 and our f of X is going to be our line. Rather, our f of why is going to be our line. So it will be two. Why minus her parabola, which is why squared minus three. Do you I and that&#x27;s it. If we had to actually solve this, we would probably used the integral with respect to why? Because it&#x27;s more straightforward. There&#x27;s no radical science, but for the purposes of this problem, we&#x27;re done

David Marsella · Answer

So here we have a region defined by why equals radical X and Y equals X cubed. And we want to express that region in terms of an integral with respect X and y. So we have Our general form is integral from a to B f of X minus G of x dx and we also have the integral from a to B of f of why minus g of y do I. So we&#x27;re giving the equations in terms of X. First, let&#x27;s get them in terms of why eso for this one will end up with X equals y squared. And for this one will have X equals the hue Groot of Why now we need to find We know that in both of these cases, when X equals zero equals zero. So this boundary point is the 0.0.0 We will need this boundary point, which is the intersection. So we&#x27;re going to set them equal to each other. Eso X cubed equals radical X Uh, which recon simplifies extra six halves equals extra 1/2 and then we can combine those two X to the five halves equals zero. So we have X equals zero and one very new zero. Uh, so it&#x27;s one and plugging that in, we get the 0.11 So these are our boundaries. So if we do with respect to X, we set this up. Our a is going to be zero r b is gonna be one R f of X is going to be our top red line cause we&#x27;re gonna do our vertical line check. So at the top of this line, we have why equals radical X so radical X minus R G of X is gonna be this lower black line, which is why equals X cubed, so X cubed DX and that&#x27;s it that&#x27;s are integral. And then respect toe. Why again are a is gonna be zero and R B is gonna be one. Now we have, ah, horizontal line the right side of a horizontal line. Hits are, uh, X equals que brood of Why so that&#x27;s our f of X. So we have the cube root of why and then our G of X is our red line. Our X equals y squared. So our g of x is y squared de y. That&#x27;s it. We&#x27;ve set up both into girls

David Marsella · Answer

Okay, so here we have a region to find by X equals the absolute value of why and X equals two minus y squared. And we want to find the area of this region both with respect, X and disrespectful. I were to do it both ways. We want to make sure that her answer matches at the end. So we know our general form. Yes, Our area is equal to the integral from A to B of F minus G. We&#x27;re gonna do it for FX, G of X and F of Weijia. A few notes We know that our X equals absolute value of why is actually essentially to separate equations. We have our X equals y in the positive quadrant and we have an X equals negative. Why in the negative quadrant, in terms of X, they&#x27;re basically the same thing we have like was X And why equals negative X Our parabola X equals two minus y squared. If we want it in terms of X, we get why equals plus or minus the absolute value of two minus X. So now we have our equations in terms of X and Y we&#x27;re gonna need dollar various points. So we&#x27;re gonna have potentially these four points. We have the two intersection points and we have to ex intersection point. No, our origin is easy. X equals wide 00 So that&#x27;s that point. Our parabola is pretty straightforward. Uh, it&#x27;s gonna be 20 The intersection points. We&#x27;ll find this one. We have X equals y and X equals two minus y squared. If we set those equal to each other, we get Why equals two minus y squared. You can rearrange it into quadratic equation, and it factors into why. Plus two, Why minus one. So we get why equals negative two and one. The negative two is off our graph. It&#x27;s not even relevant to the region we&#x27;re looking at. So are one is important. Plug it in and we get the 0.11 Now we could do that again for the bottom, but we know that both of these air symmetrical around the X axis, so we know that they&#x27;re going to have the same X value and opposing y values. So we know this is simply one negative one, and that gets us all our our parts. We have the boundaries for the intervals with our various intersection points on. We have all of our equations in terms of X Ally, Let&#x27;s get started. Let&#x27;s find our area in terms of X first. So in terms of X, we&#x27;re going to use a vertical line, Teoh, to check our regions, we&#x27;re gonna have to regions, you know, divided by the line at X equals one. So our first region is our absolute value and since it&#x27;s symmetrical, we&#x27;re just going to use our positive, are positive line and multiply it all by two. So we have to times the Inter girl from zero toe one of X. Yes, plus, her second region is just the parabola. So again, we&#x27;re just gonna use the positive value and multiply the whole thing by two so it will be one to to and radical. Tu minus X dx are integral for radical two minus X is going to require some use substitution or you is gonna be two minus X and D you is going to be negative. DX then when you resolve that are anti derivative is gonna be two times the quantity of X squared over two from 0 to 1 minus to times two times tu minus X to the three halves all over three from 1 to 2. When we substitute, we&#x27;re going to end up with two times the quantity of 1/2 minus zero minus two times the quantity of 0/3, minus 2/3. And then that should all simplify down to 7/3. Right? And now, with respect to why, with our horizontal line test, we see that our we&#x27;re gonna have to regions separated by the X axis. So our first region, our lower region, is gonna go from negative one 20 And with a horizontal line test, the right most equation is our parabolas. That&#x27;s gonna be our f of X or f of why rather tu minus y squared that will be minus negative. Why or plus Why? Because our g of X is negative. Y do I plus integral from 0 to 1 Once again, our parabola is our f of why and this time RG of y is simply why so it&#x27;s minus or g a y de y. This doesn&#x27;t require any use institution. We can just get our anti derivatives right away. So we end up with two. Why minus why cubed over three plus why squared over two from negative 120 plus to I minus y cubed over three minus y squared over to from 0 to 1. Our substitution gets us zero minus the quantity of negative two plus 1/3 plus 1/2 plus the quantity of two minus 1/3 minus 1/2 minus zero. And again, when you simplify everything down, we should come out with 7/3. So the area of this region is 7/3 and that&#x27;s it. We&#x27;re done.

Problem

Sketch the region enclosed by the curves and find…

Problem 6 Easy Difficulty

Find the area of the shaded region by (a) integrating with
respect to x and (b) integrating with respect to y.
(Graph cant copy)

Answer

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Howard Anton, Irl Bivens, Stephen Davis

Calculus Early Transcendentals

Heather Z.

Kristen K.

Michael J.

Joseph L.

Integration Review - Intro

Definite vs Indefinite

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Calculus Early Transcendentals

Heather Z.

Kristen K.

Michael J.

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Integration Review - Intro

Definite vs Indefinite

Howard Anton, Irl Bivens, Stephen DavisCalculus Early Transcendentals

Heather Z.

Kristen K.

Michael J.

Joseph L.

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Howard Anton, Irl Bivens, Stephen Davis

Calculus Early Transcendentals