Question

first time attempting this method

Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves and lines about the y axis

y=7x, y=-x/7, x=1

Answer 1

are you familiar with the shell method?

Answer 2

not at all!

Answer 3

So forget the idea of "rotating" it around the y-axis. Instead, imagine taking a bunch of cups that fit inside of one another where the center is the y-axis, kind of like this:

This is the idea behind the shell method. Each of those "cups" I've drawn are really cylinders, and what we do is by using the magic of calculus have an infinite number of cups inside of cups at varying heights that are infinitely thin, so when we add up all infinite number of them we get a regular, normal, finite number back!

So before we can add up an infinite number of infinitely thin cups, how do we find the volume of just one of those "cups"? Well, remember, volume is length*width*height. So how do we use that here? Well, what's the surface area of one of these cups? It's just a cylinder, so if you unroll one, you have

Now, at any point, we see that the height will be our function, and the length is the circumference of a circle at that value of x, so the surface area of the cylinder is: f(x)*2pi*x

But remember, this is just area and we need volume, so when I said each cylinder was infinitely thin, I MEANT it! So what do I do? I multiply it by "dx" which is an infinitely thin dimension as it's width. Now I have L*W*H=V

But (2*pi*x)*(f(x))*dx is just an infinitely thin volume at some point x. Now you must add the integral "S" to add up all these infinitely thin volumes between some point and another point, and you will have your result!

Make some guesses, show me what you can do and I'll make sure you understand. =)