Question

Suppose you want to find the volume of a random function like sin(x) rotated around the x-axis from 0 to pi. It'll be a kind of ball-like thing, but not quite a sphere, right?

Answer 1

Is this a fine example or do you want a different one?

Answer 2

I would think the messy ones are when the axis of revolution is the x or y axis or when its distant from the graph itself.

Answer 3

*isnt

Answer 4

noo i like it, start small

Answer 5

I suppose we can do whatever we need to, we can do the shell method instead, but either way once you get the idea from something simple, you'll start to see what's happening.

I'll continue with this example then for iambatman.

Answer 6

We can do like 2 more after this if you want, I don't care.

Answer 7

take my money

Answer 8

Rotation is a linear transformation.

Answer 9

Take my knowledge call it even k. Let's actually do this.

Answer 10

wait im grabbing a note book i want to actually learn this

Answer 11

good to go

Answer 12

Ok, so let's get this started, we'll say:

y=sin(x)

Now we can think of it like this, sure, but really, this is being rotated around like a circle, so it's more like:

r=sin(x)

since it's the radius of a circle that's changing depending on where you are on the x-coordinate.

Answer 13

So at any point we can see that we have: so then the area of a little circular disk with this radius that changes depending on where we are is:

A=pi*r^2 right?

So plugging in r=sin(x) we get:

\[A=\pi \sin^2(x)\]

Does this make sense so far? We basically just made a function so that Area is in terms of where we go on our x-axis.