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8 Use the method of cylinders to determine the volume of the solid obtained by rotating the regionbounded by y = J=S-Ix,T=-L and < =6 about the line x =-2 _ X+2...

Question

8 Use the method of cylinders to determine the volume of the solid obtained by rotating the regionbounded by y = J=S-Ix,T=-L and < =6 about the line x =-2 _ X+2

8 Use the method of cylinders to determine the volume of the solid obtained by rotating the region bounded by y = J=S-Ix,T=-L and < =6 about the line x =-2 _ X+2



Answers

Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis.

$ y = x^{\frac{3}{2}} $ , $ y = 8 $ , $ x = 0 $

I'm gonna be using this method for this problem. Which means I have V is pi. Times are bounds from 2 to 4 times are why? Which is negative ax squared plus six X minus eight. Now, remember, one crucial piece of information. This has to be squared. What's in parentheses? Haas To be squared times, DX. Which means now, before we take the integral, I would recommend doing the extra step of distributing and foiling to make this easier to recognize the separate terms. Otherwise, it gets a lot more difficult when you're trying to integrate. Now we can integrate. We're gonna be using the power method, which means we increased the expert it by one and then we divide by the new exponents. As you can see over here 1/5 x to the fifth. That's a good example of this. If you have a coefficient, please take that into account as well. When you're multiplying by whatever you're dividing by, if you're dividing by the new exponents plugging end, we end up with plug in four minus plugged in to via 16. Divide by 15. It's a 16 pi over 15

We know the radius is gonna be ex. The height of the cylindrical shell is negative. X squared plus X X minus eight. As specified in the problem. Let us plug into the formula for V. We have two pi times the integral from 2 to 4. Those are bounds times radius. We just at his axe times height, which has said his night of X squared plus six acts minus eight times D of ax. Okay, we're gonna be integrated when we integrate. Use the power rule, which means we increased exported by one. And then we divide by the new exponents. So remember, this was negative. X squared. When we multiplied this by acts, we end up with negative acts. Cute, which means negative X Cube becomes night of X to the fourth over four. So remember that we have to be distributing by what's on the outside. For all three of these terms, there's an algebraic concepts you probably learned years ago. Okay, now that we've got our integral, we know it's time to plug in. Now, this is a little bit long, but you basically just have to plug in your upper bound plugged in minus your lower bound plugged end minus two pod times negative to the fork over four plus two times two cubed minus four times two squared, which gives us eight pi as our solution.

First we want to know where to equals one plus y minus two. So that means that one equals one minus two squared. And so the square root of one equals two square it of why minus two squared. So we have plus or minus one equals. Why minus two, which gives us positive one equals y minus two or negative one equals y minus two. And so we get Why equals three for why equals one. So that means we're going to be integrating from 1 to 3 to pie y times two minus one plus y minus two squared de y. And if we take our to pie out of the integral, we're left with two. Why minus why the times one plus y minus two squared. And that's going to result in two y minus. Why minus y times. Why Might has two squared de y and then we're going to expand the Y minus two squared to what? To y minus wise. Why? And then minus y times. Why squared? Minus four y plus four. And then we have Why minus y cubed plus four way squared minus four y In the last week, we can simply combine this why? And minus four y to get to pie times the integral of negative y cubed plus four y squared minus three way. Okay, de y. And now we're ready to take our integral. So we have two pi times. Negative. 1/4. Why did the fourth plus 4/3? Why cute? Minus three halves. Why squared from 1 to 3? And if we plug in three, we get to pie times Negative. 1/4 times. Three to the fourth, plus 4/3 times three cute, minus three halves. I'm three squared minus. We played in one. We get negative 1/4 plus 4/3 minus three halves. And that gives us two pi times negative. 81/4 plus. Well, no. Eight over three minus 27 over to minus negative. 1/4 plus 4/3 minus three halves. And if we distribute this negative sign here and put our fractions with common denominators next to each other, we have negative. 81/4 plus 1/4. Plus when I hate over three plus 4/3. Sorry. Minus 4/3. You know what? Minus 27/2. Put 3/2. Okay. And Now we have two pi times Negative. 80/4, plus 10 for over three, minus 24 over to, and that's going to give us two pi times negative. 20 +10 for over three minus 12. Negative. 20 minus 12 is negative. 32 and we have our post. One of 4/3. Now, if we get a common denominator, we have 32 times three is 96. So negative 96 +104 over three gives us two pi times 8/3 for a final answer of 16 pie over three.

Helen and Rain. Welcome back. Okay, we're in chapter six. So with my pen be chapter six, Section three, Page 4, 44. Number 11 were given Michael's execute y equals eight. Come on and X equals zero the y axis. You'll find the region rotation. Now we want to do is rotate that around the X axis using shells. And so we need a parallel stripper rotation. This case thickness it ey the radius rotations. One over is why so we get TV Is two pi times the radius rotation times the height of the strip. Now the highest room has to be a function of Why So that would be the cube root of why eso for X here, right? And then the thicknesses in d y. All right, so here's the circumference rotation. Here's the height of the strip thickness descriptors your three dimensions. Well, I'm a benefit testament Thin strip revolving around excess function. Why? So let's integrate from what is it? 08 Hawaii. Now what the keyboard requires where one third times whether one is one of the one of the 31 of the four thirds, and so it's power rule again to V equals two pi times. All right, so what is four thirds plus one? That would be seven thirds the virus, seven thirds or multiple leadership. 3/7 evaluated from 0 to 80 drops out. So eight to the one third. Okay, Hang on to buy about six papers. Seven, three times 2.65665 or seven times a million. Zero dropped out. So it's just eight to the seven thirds. So the one third is Cuba. Debate is to to the 71. 28 uh, six times. That is 46 times 20 is 66. 1 way, six times 106 100. So let's see. Great. 16 status. 48 16. He was 20 to 1. 68 for 6. 77 68 days ago. 768. 7th of by cubic units. They go off those helpful numerator. See you next time. Bye bye. Good luck with Joe. Mark


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