# Volume of revolution $y = x^2 + 2$

$$y = x^2 + 2, 0 \leq x \leq 2$$

I know what this looks like, I was suppose to find it for a revolution around the x axis for which I used the disk method and around the y axis which I used shell so I never had to change variables.

I got two different numbers after about an hour of calculations, that is not possible right? Just to make a logical check if I take my calculator and spin it in any way it has the same volume whether I spin it left or right.

Is it possible that the shape has different volumes depending on the axis?

X-axis revolution (Pretend that it's filled in):

Y-axis revolution:

In this case, the X-axis is directed to the right and slightly out of the screen, and the Y-axis goes into the screen and to the right at about a 45-degree angle.

It is absolutely possible you'd get two different volumes if you rotated it about 2 different axis... think of extreme cases like y=.001x . If you revolved that around the x axis from [0,2], you're going to get a much different volume than if you rotated it around the y axis from [0,2].

You may get different volumes if you rotate around different axes. Imagine rotating your calculator along its shorter axis (you get one cylinder), and then imagine rotating it around its longer axis (you get another cylinder).

You will get different numbers depending on which axis you rotate around.

Think about the line $y=2x$, where $0 \le x \le 1$.

If we rotate around the $x$-axis we get a cone with height one and base radius two. The volume is: $$V = \tfrac{1}{3}\pi r^2h = \tfrac{4}{3}\pi$$

If we rotate around the $y$ axis, we get a cone with height two and base radius one. The volume is: $$V = \tfrac{1}{3}\pi r^2h = \tfrac{2}{3}\pi$$

The volume when you rotate about one axis is hardly ever the same as the volume when you rotate about another axis.

So there is absolutely no reason for your two answers to be the same.

As a simple example, rotate the region below $y=1$, above the $x$-axis, from $x=0$ to $x=3$, about the $x$-axis. You get a cylinder, radius $1$, height $3$. The volume is $3\pi$.

Now rotate about the $y$-axis. We get a fat cylinder, radius $3$, height $1$, volume $9\pi$.