$y = x+\sin(x)$ Solid of Revolution y-axis with implicit function.

Question

Given the equation $y = x + \sin(x)$, as far as I am aware, an explicit equation $x = f(y)$ can not be found. Is there still a way to compute the volume of the solid of revolution of this function rotated about the y-axis, because the volume of the solid of revolution about the x-axis can be found ($V = \pi \int (x + \sin(x))^2 dx$). Considering $x = f(y)$ this "function" appears to have points with infinite slope on integer multiples of the point $(\pi,\pi)$ which may also complicate the computation of the solid of revolution.

Graph of $y = x + \sin(x)$

Answer 1

You've not specified a bounded region or interval over which to compute the volume of revolution, but in general, we can use the method of cylindrical shells to compute the volume when the cross-section is rotated about the $y$-axis. So if $f$ does not have a closed-form inverse, we can still compute the volume as

$$V_1(a,b) = \int_{x=a}^b 2\pi x f(x) \, dx,$$ for some suitable interval satisfying $0 \le a < b$, whereas the volume of the region rotated about the $x$-axis is given by the method of disks/washers; e.g.,

$$V_2(a,b) = \int_{x=a}^b \pi f(x)^2 \, dx$$ where $f \ge 0$ on $[a,b]$.