I'm trying to explain how you would find the volume of rotation of $y=x^2$ around $y=19$. I am aware you can just translate the graph downwards by $19$ but the point is to try explain volumes of rotation that aren't around the axis with a simple example. My explanation is as follows:
To find the volume of rotation, first divide up the curve $y=x^2$ within the domain $x\in{\left[-\sqrt{19},\sqrt{19}\right]}$ into $n$ blackened subsections like the one in the diagram such that $\Delta x=\frac{2\sqrt{19}}{n}$. To approximate, we assume each section when rotated about $y=19$ creates a cylindrical disc where $r_i$ is the radius of the base of the $i$th disc. From the diagram, we can see that $r_i=19-f(x_i)$ where $f(x)=x^2$ and $f(x_i)$ is $f(x)$ evaluated at the $i$th subsection. Therefore we have the volume, $D_i$, of one disc is $D_i=\pi \left(19-f(x_i)\right)^2\Delta x$. As $n$ increases i.e as $\Delta x$ decreases, $D_i$ better approximates the actual volume of the truncated cone formed when the black section is rotated about $y=19$. Therefore the desired volume of rotation, $V$, is given by $\displaystyle V=\lim_{\Delta x\rightarrow0}\sum^{n}_{i=1}\pi \big(19-f(x_i)\big)^2\Delta x$. By the fundamental theorem of calculus this means that $\displaystyle V=\int_{-\sqrt{19}}^{\sqrt{19}}\pi\Big(19-x^2\Big)^2\;dx$.
What I want to ask is that does this explanation, make sense in a rigorous mathematical sense. I am particularly unsure about the line " By the fundamental theorem of calculus this means $\displaystyle V=\int_{-\sqrt{19}}^{\sqrt{19}}\pi\Big(19-x^2\Big)^2\;dx$. " That was the best I could do to justify the jump from a summation to an integral. I'm not aiming to rederive the whole process using the first principles formula but I just wanted to know, can I quote the fundamental theorem of calculus like that?