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Kemaining 0Z 08 48 Submit TestThis Question: pts18 0/ 25 complele}This Test; 100 pls possibleLet Rne the regilon bounded by the following curves Find the volume sho...

Question

Kemaining 0Z 08 48 Submit TestThis Question: pts18 0/ 25 complele}This Test; 100 pls possibleLet Rne the regilon bounded by the following curves Find the volume shown lo Ihe righl aboul Ihe X-axtsgererated revotvirg the shaded region6x Y= and *integral that gives Il volumc 0l Ine solidJ Oa (Type exact answcos ) Tne volume 0l Ine sotd (Type adCdMstdcubic untsEntef Your dns CiRach7nsin Doxc:F[SLs

Kemaining 0Z 08 48 Submit Test This Question: pts 18 0/ 25 complele} This Test; 100 pls possible Let Rne the regilon bounded by the following curves Find the volume shown lo Ihe righl aboul Ihe X-axts gererated revotvirg the shaded region 6x Y= and * integral that gives Il volumc 0l Ine solid J Oa (Type exact answcos ) Tne volume 0l Ine sotd (Type adCdMstd cubic unts Entef Your dns Ci Rach 7nsin Doxc: F[SLs



Answers

$33-38=$ The region bounded by the given curves is rotated
about the specified axis. Find the volume of the resulting solid
by any method.
$$y=-x^{2}+6 x-8, y=0 ; \quad \text { about the } x$$

Were given a set of curves and in line and we're asking that an integral for the volume of the solid obtained by rotating this region bounded by thes curves about this line the curves are y equals e to the negative X squared y equals zero X equals negative one and X equals one in part A The line that were rotating around is the X axis. So for this problem, I think might be good to sketch the region first. Yeah, so function y equals e to the negative. X looks something like this. See that it changes from con cave down to con cave upward at some point. Yeah, So this is the red region. We want to rotate it around the X axis, which is the dashed line. And if we do so, what we obtain is a solid with cross sections which are disks. Therefore, the volume of the solid is the integral from X equals negative one positive one of the areas of thes disks which is pi times the radius of the disk which is each the negative x squared, squared DX. This is pi times and because our function is even two pi integral from 01 of each of the negative two x squared the X and using a calculator. Yeah, this is approximately 3.75825 in Part B. Yeah, the line that were rotating around is the line y equals negative one. So now, instead of the salt being with cross sections which are discs, cross sections which a parallel to the y axis our washers. And so the volume is the integral from Mexico's negative one positive one of the areas of these washers, which is pi times the larger radius, which is top function Eat the negative X squared minus the bottom function negative one squared minus thes smaller radius. Yeah, this is y equals zero top function minus y equals negative one The bottom function squared dx and this simplifies to mhm. For one thing, knows that the function is even inside the instagram. So this becomes two pi times the integral from 0 to 1 of and then we have each of the negative two x squared plus two e to the negative X squared DX and plugs into a calculator. This is approximately 13.14312

If we consider the rejection on the facts. Picture we rotate about. Why? So we would say it About why equal to one. Then this gives you the second image. So the inner radios. So Inna radios, It's equal to zero. And our out radios, It's equal to 1 -1 you eggs to a cross sectional area. E. X. Is going to be pie. You have won my nets son Q. X squared minutes By zero square. And this will give us By one. My next son cube eggs suede. So we find the intersection points of why? So intersection intersection coins of way equal to turn Hugh eggs aimed Why equal to one. So then this implies that Son Cube X will be equal to one. So then it implies that son Eggs equal to one. So our S. Uh x. Therefore will be equal to buy divided by four. So therefore a solid lies to the solid. The solid lies between X. equal to zero and by on four. So then we can fight them volume. So a volume V. I would therefore be equal to the X. Eager from several. Too bye on four you have E. X. The eggs. And this is equal to the insignia 02 by on four. You have by one minus son. You X squared Dean X. As the final answer of our integration. So this is the integra yes. Setting up

Were given a set of curves and a line and were asked to set up an integral for the volume of the solid obtained by rotating the region bounded by these curves about this line and then to use a calculator to evaluate this integral, correct to five decimal places the curves air y equals X squared X squared plus y squared equals one. And why equals zero are, in fact, why is greater than or equal to zero? In part A. The line is Thea X axis. So before I do any calculations, first I'll just sketch a graph of this region. So this is going to be on Lee in the first two quadrants here. So the problem y equals X squared. His 0.0 11 negative 11 has a shape like this X squared plus y screen was one. Is this circle so really destroying a semi circle with a radius of one centered at the origin? And so this region here in red is the region that we're interested in rotating. It would be informative to find where it is that this these points of intersection of these two curves, so to do this. We'll have to solve this by plugging in y equals X squared. So we get that. Why, plus why squared is equal to one. Or in other words, why squared? Plus why minus one equals zero. This implies that why is equal to or maybe the easier way is to find the X value first. That's what we're interested in for this part anyways, instead of finding it that way, since why is equal to X squared? We have that X squared plus X to the fourth is equal to one or that X to the fourth plus X squared minus one equals zero. And then, using quadratic formula, we have that X squared is equal to negative one plus or minus the square root of five over to. But because X squared is to be a positive number, this has to be the positive square root of five and noticed that for grafting purposes, this is about 0.618 which okay, and then from this it follows that X is going to be plus or minus some constant, which will call a which is really plus or minus the square root of negative one plus 35 over to and using a calculator again. This is approximately plus or minus 0.786 And we see that this does match your graph pretty well. So we have that are volume looking at our solid. If you rotate around the X axis, the cross sections air parallel to the Y axis and our washers. So the volume is the integral from what we've conveniently labeled as X equals negative a two X equals positive A of the area of these washers, which is pi times. Then we have the outer radius. This is the top function, which is why equals square root of one minus X squared minus the bottom function which is simply X equals zero squared minus and then the inner radius. This is the bottom function, which is Michael's X squared squared T X notice that this function in the Inter Grand is even so this is the same as two pi times the integral from zero to positive a of in simplifying, we get one minus X squared minus X to the fourth DX. And if we plug this into a calculator, he's in the value of a we found previously this is approximately 3.54459 in part B. The line is about the Y axis. Oh, we have simple shortcut here for finding the Y coordinates. But these intersections, if the X coordinates air plus or minus A and the why coordinates are going to be a squared. So if you rotate around the Y axis once again our cross sections, they're going to be parallel to the y axis. Probably the easiest way to evaluate, actually. Oh, but differently, cross sections are parallel to the x axis, and we'll actually have two distinct regions. So the region above the line y equals a squared in the region below. Yeah, And so the volume is going to be the volume of spot, um, part of the solid, which is the integral from y equals zero to a squared, and then this solid. After being rotated, the cross sections two parallel to the X axis are going to be disks. So this is times the area of the disks, which is pi times the radius, which uh huh is going to be X equals the square root of why squared d y. And then we're also going to add the volume from the top part of the solid, which is from why he was a squared to Why equals one and this part of the solid when rotated cross sections air parallel to the X axis and our disks. So this is integral of the area of these discs, which is pi times the radius of the disks, which we can see a rearranging our equation for the circle. This is X equals positive square roots of one minus y squared squared de y simplifying. This is pi times the integral from zero to a squared of Why D Y, plus high times the integral from a squared to one of one minus y squared de y and plugging thes into rolls into a calculator. This is approximately 0.99998

First, we need to find our bounds of integration so we'll do that by setting negative X squared plus six X minus eight. You go to zero, and if we factor out a negative one, we're left with X squared minus six X plus eight. And now if we factor X squared minus six. Expose eight. We get X minus four times X minus two, which gives us X equals four and X equals two. So we're integrating from 2 to 4 and we have two pi times the radius, which is X times negative X squared plus six X minus eight. Now, if we distribute the X, we have too high times the integral from 2 to 4 of negative execute plus six X squared minus eight X And now we're ready to integrate, so we'll get to pie times Negative. 1/4 x to the fourth plus two x cubed minus four X squared. Evaluated from 2 to 4. And if we put in four and two, we're going to get to pie times. Negative. 1/4 times for the fourth, plus two times for cubed minus four times four squared minus negative. 1/4 times to the fourth plus two times two cubed minus four times two squared. And that's going to simplify, too. Two pi times. Let's see if four to the fourth, divided by fours for cubed. So we have negative. 64 plus 1 28 minus 64 right? My is to the fourth is 16 to bag by four. Negative for plus two times two Cubed is to to the fourth, so 16 minus four times two squared, which is 16. So we get negative. 64 minus 64 Negative. 1 28 plus 1 28 gives us zero and then we get 16 minus 16 here, which gives us zero. So we really have to pie times a negative negative four for two pi times four, which gives us the volume of a pie.


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