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Numerade Educator

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Problem 3 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 = e^{-x^2} $ , $ -1 \le x \le 1 $

Answer

a) (i) $$\quad S_{x}=\int_{-1}^{1} 2 \pi e^{-x^{2}} \sqrt{1+4 x^{2} e^{-2 x^{2}}} d x$$
(ii) $$\quad S_{y}=\int_{0}^{1} 2 \pi x \sqrt{1+4 x^{2} e^{-2 x^{2}}} d x$$
b) (i) $$11.0753$$
(ii) $$3.9603$$

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Video Transcript

you want to have this function? Um, exploit plane? No. The function why is hard to even minus X squared Bonzo Verizon increasing portion here. But it is symmetric is otherwise the X squared. So and that actually want to rotate this about the X axis. So from minus one upto one security that we're gonna do is gonna obtain something that looks like like a It's like a funny, uh, like a camp, like a some special. So, like these this one shape And, uh, that's you attend the service you're gonna do. Compute. What is the area off these? Uh, yeah, on the surface. Obtained by routine about the x axis. So the area here will be equal toe interval from minus one off to one off to five f bricks square the one plus if Prime sward. Yes, his name. Okay. Is that the length of the curve differential of length of the curve? Sbx on. Uh well for our function or function is eyes that so perfect is able to read to the minus X squared so that if you have the computer do it, it would be for the movement. Exponential is the exponential. I mean by the channel would have video for this which is minus toe X sort of the square square that Brian squared there will be able to that square which is four x squared on this score is bad. I'm sled the voices their exponents ad would be four next square times e didn't minus two x squared So that this in the role this area is equal toe Yeah, to find they drove from minus one upto one off to buy them. Saw our function You can minus X squared times the square root one plus, uh, if Prime square which is that for X squared? McGrann my school x squared. Yes, Onda way. Want to rotate now? That But, uh is of the curve. Uh, from well, from from my is one of 21 She got this girl from minus one after one. What were stated about the y axis? So these girls like these. We're datable the y axis. You have these the shape like God, like a genius hat. But so since this name is because you couldn't see that all the is part you could be like This shape is the same. A city which is rotated this length from zero up to one. Because if we consider from when is one or 21 we would be counting twice. So the more for the area. So the area here you are obese, but they remember the Y axis is gonna be equal toe the interval from zero up to one, 25 on the while x my ex time. So our is the X, which is the same function. It's called one plus four x squared. I'm sitting minus two x squared. All right, organize side of this world and the X on the or the approximate blood with our favorite cleaner. This is the 11 point seven. So she from the this one is three 0.9 603 So those are hard to let saw the tension a bulge, the y axis. I mean, no additional. The Y axis is his vision on the rotation about the X axes. So it's decision of from 0 to 1. Textiles, that is quite a bit. And then things one is rotation about Yeah, x axis, X axis