The solid generated by rotating $y=\sin x$ on $[0,\pi ]$ has volume ${\pi }^2\mathrm{/}2$

Given: The profile $f(x)=\sin x$ on $[0,\pi ]$ and its solid of revolution about the $x$-axis.

If $a\le b$ and $f:[a,b]\to [0,\mathrm{\infty })$ is continuous, then its solid of revolution about the $x$-axis is compact and Jordan measurable and has volume $\pi {\int }_a^{b}f(x{)}^{2}dx$ (The disc formula for the volume of a solid of revolution).

$\sin \pi =0$, and $\sin x>0$ for every $x$ with $0<x<\pi$; thus $\pi$ is the first positive zero of sine, in particular $\pi >0$ (Pi is the first positive zero of sine).

The functions $\sin$ and $\cos$ are differentiable on $\mathbb{R}$, with $(\sin x{)}^{\mathrm{\prime }}=\cos x$ and $(\cos x{)}^{\mathrm{\prime }}=-\sin x$; also $\sin 0=0$ and $\cos 0=1$ (The derivatives of sine and cosine are cosine and minus sine).

For a real-valued function on $A\subseteq \mathbb{R}$, differentiability at a limit point $c\in A$ implies continuity there (A function differentiable at $c$ is continuous at $c$).

For every real $x$, ${\sin }^{2}x=(1-\cos 2x)\mathrm{/}2$ (Double-angle and quadratic power-reduction identities).

Every continuous real-valued function on a nondegenerate closed interval is Riemann integrable there (A continuous function on $[a,b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion).

The integral is linear on integrable functions (Integrable functions on $[a,b]$ form a set closed under sums and scalar multiples, and ${\int }_a^b(\lambda f+\mu g)=\lambda {\int }_a^{b}f+\mu {\int }_a^{b}g$).

If $a<b$, every constant function on $[a,b]$ is integrable and ${\int }_a^{b}c=c(b-a)$ (If $m\le f\le M$ on $[a,b]$ then $m(b-a)\le L(f,P)\le \underset{‾}{{\int }_a^b}f\le \overline{{\int }_a^b}f\le U(f,P)\le M(b-a)$ for every partition $P$; in particular every constant function is integrable, with ${\int }_a^{b}c=c(b-a)$).

If $a<b$, $G:[a,b]\to \mathbb{R}$ is differentiable at every point of $[a,b]$, and $G^{\mathrm{\prime }}$ is integrable, then ${\int }_a^b{G}^{\mathrm{\prime }}=G(b)-G(a)$ (The second fundamental theorem: if $G$ is differentiable on $[a,b]$ with $G^{\mathrm{\prime }}=f$ and $f$ is integrable, then ${\int }_a^{b}f=G(b)-G(a)$).

If real functions $g$ and $h$ are differentiable at the relevant points, then $(g\circ h{)}^{\mathrm{\prime }}=(g^{\mathrm{\prime }}\circ h)h^{\mathrm{\prime }}$ (The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g(c)$, then $f\circ g$ is differentiable at $c$ with $(f\circ g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(g(c))g^{\mathrm{\prime }}(c)$).

Sums and scalar multiples obey the usual derivative rules (Sums, scalar multiples, products and quotients: $(f+g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)+g^{\mathrm{\prime }}(c)$, $(\alpha f{)}^{\mathrm{\prime }}(c)=\alpha f^{\mathrm{\prime }}(c)$, $(fg{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)g(c)+f(c)g^{\mathrm{\prime }}(c)$, and $(f\mathrm{/}g{)}^{\mathrm{\prime }}(c)=(f^{\mathrm{\prime }}(c)g(c)-f(c)g^{\mathrm{\prime }}(c))\mathrm{/}g(c{)}^2$ when $g(c)\mathrm{\ne }0$).

Both sine and cosine have period $2\pi$ (The zero sets of sine and cosine and the least positive common period 2 pi).