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Find the surface area of the figure generated by revolving x = 6 cos(t), Y = 6 sin(t) where 0 < t < 5/4 about the Y-axis:...

Question

Find the surface area of the figure generated by revolving x = 6 cos(t), Y = 6 sin(t) where 0 < t < 5/4 about the Y-axis:

Find the surface area of the figure generated by revolving x = 6 cos(t), Y = 6 sin(t) where 0 < t < 5/4 about the Y-axis:



Answers

Find the area of the surface generated by revolving the given curve about the $x$ -axis. $$y=6 x, 0 \leq x \leq 1$$

Hello, everyone. Today we're going to solve problem number 14 here equal toe toe by another 0 to 6. Why off X Hindu Group off one plus Where does X The whole square? The X, which is equal to to buy in double 0 to 6. Text by two in the room, five by two de it's which is equal to Route five by 2 500 0 to 6 X dx, which can be given us root fight by to buy in the X squared by two. Limited from 0 to 6, which is a good toe. 55 by four six square minus zero square with physical to nine. Route five by Thank you.

So here we're going to find the area of the surface generated by rotating this curve about the X axis Between the bounds of x equals one, x equals three. And to do that we're gonna be using our area of a the surface area of a revolution solid or revolved curve however you want to say it. And so first thing we're gonna need is to get our derivative in terms of X. So we're going to use some quotient rule here. This is gonna be derivative over the top, times the bottom minus diverted at the bottom minus the directive at the bottom minus the top, all over the bottom squared. So let's do some simplifying. We got 48 Next to the seven minus 16, Next to the seven minus 32 X. All over 64 x to the fore. They fixed this for it's bugging me. So let's go ahead and take out a 32 from top and the bottom. After we subtract these two, We get next to the 7 -1 Over two X. of the four. So here's what we're gonna use for our area integral. So don't quite have enough space. I just go ahead and start down here. So we're gonna take the integral from 1 to 3 times our why? Or are ffx next to the six plus 2/8 X. Squared. And now we're gonna plug in our Square root of one plus the derivative squared Next to 7 -1 Over two X. of the four. And it's going to be dx. All right. So the first thing I'm gonna do is expand this inside. Ah Let's do the whole thing once. I'm gonna expand this entire inside the radical and see what I can get. So let's put a star here and the star here. So, if we expand the X seven minus X over two X. To the forest squared, We'll get next to the 14 -2 x. plus X squared. All over four actually eight. So this is going to be and then we have a plus one. So I'm gonna go ahead and change this into for extra eight Over four weeks of the 8 to get a common denominator. And then this will add with our middle term and then in the numerator, So these two ad And so -2-plus 4 will give us a positive too. So this becomes Next to the 14 Plus two extra 8 plus X squared All over four times next to it. And so this numerator will now factor in the same way it did before in our original ah area integral. But this time it will have a positive second term. So finally we can write this ass Next to the seven plus x. All over to exit before all squared. And so this is what is inside our square root. And if you'd like you can foil this out to sea, you'll get the same result. So let's continue with our integral. And you go from 1 to 3 Next the 6-plus 2 Over eight x squared times. Now this is what we have here. I'm going to go ahead and cancel this radical. Going to cancel this radical with our square. So what we're left with is simply what's inside the square here Next to the seven plus x. Over two x. 2 four D. X. Now let's multiply this out. Next to the six times extra seven. Give me an extra 13 Plus two extra 7 Plus X to the seven plus two X. This is all over 16 extra six. The X. Go ahead and combine these middle terms without rewriting the entire thing. This will be three X 27. And so we can just Distribute the 16 X to the six between all the terms. Oh I forgot my Sorry about that. I dropped my two pi along the way. We put this back in. Remember in our formula we have a two pi out front. So now we're gonna get a bunch of Broken up terms. So we have to pay items integral from 1-3. This is going to be extra seven over 16 plus three x over 16 plus 1/8 X. The five. This is all dX. So let's go ahead and evaluate are integral. So integral of X to 7/16. What's going to be add one the power and then divide by the new power. So it's gonna be over one, Right, 80 plus 48. Yeah And then integral of three x. over 16. It's going to be three X squared over 32. And now this is actually 1/8 X. to the -5. So we add one power and then divide by the new power, Add 1 to the power, divide by the new power. So we have 1/8. Next the -4 over negative four. And you can rewrite this as 1/32 times x to the fore the negative sign out front just like that. I forgot my evaluation bar From 1- three. So let's do some plugging in. Ah We will have 3 to the eight which I cannot evaluate in my head. Plus three squared is nine times 3 is 27 Minour through the 4th which is 81. 32 times 81. Let's just say there is that put in some more brackets, this is minus. We have ones everywhere, which makes life a lot easier. Plus 3/32 minus 1/32. And so you can evaluate this all out And don't forget to multiply by two pi at the end. You should get, we have an exact answer. You will get exactly 8429 over 81. Hi, there you are.

Problem number. Well, extra boy is equal to burn off steam minus y If dash boy is people toe 1/2 16 minus y so s is equal to bar immigration from zero Dean. It's where steam minus ride It's were one plus one over to where? What? 16 miners? Why? It's what you want. She is equipped with two point 16 minus line, one plus one over for 16 minus Roy you want which is equal to a boy? Zero? Did you? You get to corduroy lost 65 over. So right, let's let you is equal to negative for Roy 65 to be you different negative or Iran where you bonds between 65 and fine So the integration is equal toe by immigration. Uh, room 65 5 over negative half. It's wearing off you over for you, which is equal through boy one over. Will You brought three over in 65 for five after this of institutions, the final answer is liquid to 5/6 square up to fight 13 where they took 30 minus one

So we're looking at the function. Why equals six times exponential to the negative two X. And so this curve looks something like this so forever. To revolve this around the y axis, we would end up with a shape like this, not dealing with anything that crosses there. And so it's going to be this pointed sort of shape going from negative Infinity equals X. All the way to infinity equals X so to solve. This is we're rotating around the Y axis. We're going to want to use the shell method. So the formula for using the shell method is the volume equals two tens pi times the integral of x times the function of why, with respect X on this will give us infinitely many, infinitely thin shells or cylinders going all the way around that are going to help us find the full solution for this for this volume, and we're only going to deal in one directions. We're going to go from zero to infinity for X, and so that's why we're multiplying by two. Because we're going because it's going to come up with the same answer in both directions because this is a symmetrical function. So we'll have substituting everything in and adding in our bounds will have the volume equals two times pi times the integral from zero to infinity of X times six times exponential negative two x with respect to X so we can't go all the way up to infinity. So we're going to have to write this as the limits of the limit as T is approaching infinity off the integral from zero to t of X times six times exponential negative two x with respect text. So now we can solve this integral. And I'm going to use integration by parts for this because we have an ex being multiplied by an exponential function function that includes an X. So we're since we're multiplying these together, we're gonna use integration by parts setting you equal to X and Devi equal to six x nine Exponential Negative two x with respect to X So now the derivative of you is equal to one. The X and the anti derivative of D V is equal to six times exponential Negative two x divided by the derivative of this exponents so divided by a negative too so plaguing this in we're going to end up with U times V or the limit as T is approaching infinity of u times v so x Times negative three Exponential negative two X minus The integral of v times to you So negative three Exponential negative two x with respect to X and so this is going to be evaluated from zero to t and this going to be evaluated from zero to t. So now we're going to take the anti derivative of this integral. So that will end up with the limit as T is approaching infinity of negative X times three times exponential negative two x plus three times exponential negative two x divided by the derivative of negative two x so divided by negative two and this is all going to be evaluated from zero to t. So we're going to plug in tea and zero years will have equals the limit as T is approaching infinity of negative t negative three tee times exponential negative. Let's put this as a fraction so that we don't have to deal with negative exponents. So negative three tee times exponential off to T plus three negative 3/2 times exponential to t again lowering that exponents so we don't have to deal with negative exponents minus negative zero times three over exponential negative to time, zero plus three times exponential, negative to time, zero over a negative too. So now when we're evaluating this as T is getting further of further to infinity, this denominators getting larger and larger, causing the fraction itself to get smaller and smaller so that no matter how large this numerator gets this the frack the denominator is going to cause this too converged to zero next for the second term again, as T gets closer to infinity, the denominator gets larger and larger, bringing the fraction itself closer to infinity. So we'll have zero minus zero plus zero times three is equal to zero minus three times one as exponential of zero is one eso three times one over Negative too. So we're going to have this equal to 3/2 and we can't forget to multiply this all. So we have V equals two pi times 3/2 because this 3/2 is the solution for this integral. So the volume will be equal to three times pi For this for this sort of triangular code,


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