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Numerade Educator

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Problem 2 Easy Difficulty

(a) Set up an integral for the area of the surface obtained by rotating the curve about (i) the x-axis and (ii) the y-axis.
(b) Use the numerical integration capability of a calculator to evaluate the surface areas correct to four decimal places.

$ y = x^{-2} $ , $ 1 \le x \le 2 $

Answer

a) (i)$$\quad S=\int_{1}^{2} 2 \pi\left(x^{-2}\right) \sqrt{1+4 x^{-6}} d x=\int_{1 / 4}^{1} 2 \pi y \sqrt{1+\frac{1}{4} y^{-3}} d y$$
(ii)$$\quad S=\int_{1}^{2} 2 \pi x \sqrt{1+4 x^{-6}} d x=\int_{1 / 4}^{1} 2 \pi \frac{1}{\sqrt{y}} \sqrt{1+\frac{1}{4} y^{-3}} d y$$
b) $$
\begin{array}{ll}{\text { (i) } 4.4567} & {\text { (ii) } 11.7299}\end{array}
$$

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Video Transcript

So you have, ERM, now this curved the X y plane. Um, so the curve is some one of the x squared. Also, this piece, why is equal to one over X squared, and then, uh oh are extra. The minor store is the same. So not find the area by taking this skirt from one opted to So we do the surface of revolution. Well, first of all, the exactly Securitate that we're gonna obtain something like these. Sort of like a like that. Like, uh, cut cold. Something like this. So Well, we rotate above the x axis. Yeah, this service, you know, to get what is the area area. Um, there you got so well, the formula. Is that the interval from one up to two off our function. Fax What? Oh, that is there is the s there difference of little length differential of length of hair, the service, this DS time sex. So these DS would be, he says would be one plus fried squared. And then we would also want toe find what is the area if you're a date That about the y axis city of that? That seemed piece of the curve And if you rotator about the y axis, we would have something like this shaped like that. Um, see that a wealthy area in this case would be just, um, need to go from one to off X. I'm sorry, this s e X. So one plus a prime, uh, squared. Yes. Eso Let's call these one warm about money two. So for our function, our function is equal to me. Extra, minus two, So that this prime is about the minus two extra minus thief Monday. Now prime squared. It's gonna be all the ministries choirs or And this will be excellent minus six. So that our integral one these would be Yeah. The interval on I'm a factor of Dubai. Don't forget to part was missing that to fight. So one would be would be to buy sometimes our function that is extra minus two, uh, Times Square road off one last four, thanks to the minus six. Yes, from one after two. This is approximately four point 45 67 and they well for two Patijn about the way thinking about while. So these really salutation Vote x for the rotation of old. Why, we would have this needle from one what to do to buy time. Sex? I'm some these? Yes, this party's STX. That is the differential off length Be off the curve. Uh, which is that this is approximately 11 0.7 to my life. So those are or areas of revolution? Yes. Well, grab Bush. Yeah. Uh, that