Why does the washer method not apply when rotating a function along the $y$-axis

Question

I have the following region bounded by the following functions

$$y=4-x^2$$ $$x=0$$

Now, I am asked to rotate this region along the line $x = 3$ and find the volume of the solid created. I have been using the washer method to obtain the volume of shapes rotated along the $x$. However, now that I am rotating along the $y$ ($x = 3$), I understand I must switch my equations to be with respect to $y$.

However, can someone explain why I can't use the washer method along the $y$? Is it because when I get the cross-sections it is not a circle?

Answer 1

You could use the washer method, but you would have to invert the function. Visualize each washer; the distance from $x = 3$ is a function of $y$, but it's more natural to give $y$ as a function of $x$. Instead, if you use the shell method, you are adding the cylinder circumference with radius $3-x$ for $x$ from $2$ to $3$ and height $y = 4-x^2$. That's a much more natural integral to set up.

In general, the pattern to look for is to integrate along the variable which is the input of the given function. Otherwise, you may have to deal with the unpleasantness of "inverse function land". Which is okay for $y = 4-x^2$, but less pleasant for $y = \sin(x^3)$.

Answer 2

You can apply the method, in this case the set up should be

$$\int_0^4 \pi(3^2-(3-\sqrt{y-4})^2)dy$$