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(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.

$ x = y + y^3 $ , $ 0 \le y \le 1 $

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a) (i)$$\quad S_{x}=\int_{0}^{1} 2 \pi y \sqrt{1+\left(1+3 y^{2}\right)^{2}} d y$$(ii) $$\quad S_{y}=\int_{0}^{1} 2 \pi\left(y+y^{3}\right) \sqrt{1+\left(1+3 y^{2}\right)^{2}} d y$$b) (i) $$8.5302$$(ii) $$13.5134$$

Calculus 2 / BC

Chapter 8

Further Applications of Integration

Section 2

Area of a Surface of Revolution

Applications of Integration

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So you have dysfunction, your ex functional off? Why Mexico? Why, That's a white cube. So he throw that here with this one off the exact schism? Well, that when I was the way, we're gonna have a function that looks sort of like that. Um, this is a graph. The cover off accessible toe white plus y cube. You want to, But the computer is the area off. What happens if we were date? Um, this piece of the curve from 0 to 1. Well, first of all, the well, the wire about the exact cyst. So we rotate that about the exact is we're gonna happen. Soul shaped like this, um, taking that around there. Sort of like this cup. Um, Rebate about the X axes. Andi, Uh, you will also like to complete. Yeah, yeah, off that curb. But now, rotating about the y axis. So well, they could Something like that. Yeah, something like these. And then we will take Now. That's him. Interval from zero. That same piece from here with one is why excellent. Our deed y axis. Um, there are things like this shape. Mhm. So, uh, so we'll look completely serious. Yeah, right? Yeah. Here, also here, Uh, here is we have X as a function of why, so that this area is gonna be the interval from zero after one. Off. Off. Why? Um um Yes. Off the B s d y. This is the differential off length off the Yes, Hawaii on the curve, that is. Well, he's the following off. Why have to buy that? Because, uh, or your date, he's like, uh, taking like the piece off cylinder. And then that was the length. Here. WP one plus if prime Why? If you're the witness. But why di squared and see why. Right on the in for this one since, uh, for the left here, this business is gonna be why, at each point, so the area for this one would be in jail from zero off one five. Why times, Uh, data iss e y. That. This is ableto one plus v a t function squared, uh, square with the one plus a functional squared. Why? So you have that then offer dysfunction. This is why access culture. Why, as prime is gonna be ableto that this is just one plus three y squared, so that, uh, this internal over here would be in jail from zero up to one bye. Why? And then times this quoth one plus want lis c y squared. And then we have to square that So that that's why on, um, these you still there? One. This other area would be the injury from zero off one or two pi. It was the function that is it's why plus y que by them is life plus y cube. I'm sorry. Square rolled off one plus Well, bless to why squared on the that squared. Why? Yeah, And of the blotter Silver talk later is number. Here it's called Is 11 One is approximately, um 8.532 And if we go, this one's to listen to go to so too is some approximately Yeah, 13 point 51 34 So those air the area's off is to surfaces of revolution all the x axis. So this one is servicer revolution. About the X axis on these one is surface of revolution about the y axis

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