The surface generated by rotating $y=\sin x$ on $[0,\pi ]$ has area $2\pi (\sqrt{2}+\mathrm{arsinh}1)$

Given: The radius function $r(x)=\sin x$ on $[0,\pi ]$ and the surface obtained by rotating its graph about the $x$-axis.

Under the hypotheses of the scalar surface-integral theorem for a surface of revolution, the rotated surface has area $2\pi {\int }_a^{b}r(s)\sqrt{1+r^{\mathrm{\prime }}(s{)}^2}ds$ (The surface of revolution has area $2\pi {\int }_a^{b}r(s)\sqrt{1+r^{\mathrm{\prime }}(s{)}^2}ds$).

Those hypotheses require $a<b$ and $r:[a,b]\to [0,\mathrm{\infty })$ to be $C^1$ on a neighbourhood of $[a,b]$, positive on $(a,b)$, and allowed to vanish only at the endpoints (Scalar surface integrals on a surface 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 $c$).

The function $\sinh :\mathbb{R}\to \mathbb{R}$ is odd, strictly increasing, and onto; $\cosh$ is positive; ${\cosh }^{2}v-{\sinh }^{2}v=1$; and $(\sinh v{)}^{\mathrm{\prime }}=\cosh v$ (Addition formulas, identities, parity, and derivatives of the hyperbolic functions).

Let $I\subseteq \mathbb{R}$ be order-convex with at least two elements, let $f:I\to \mathbb{R}$ be continuous and injective, and let $g:f[I]\to I$ be its inverse. If $f$ is differentiable at $c\in I$ with $f^{\mathrm{\prime }}(c)\mathrm{\ne }0$, then $g$ is differentiable at $f(c)$ and $g^{\mathrm{\prime }}(f(c))=1\mathrm{/}{f}^{\mathrm{\prime }}(c)$ (Derivative of an inverse: if $f$ is continuous and injective on a nondegenerate interval $I$ and differentiable at $c\in I$ with $f^{\mathrm{\prime }}(c)\mathrm{\ne }0$, then the inverse $g$ is differentiable at $f(c)$ with $g^{\mathrm{\prime }}(f(c))=1\mathrm{/}{f}^{\mathrm{\prime }}(c)$; and if $f^{\mathrm{\prime }}(c)=0$ then $g$ is not differentiable at $f(c)$).

For every real $u$, $\mathrm{arsinh}u=\log (u+\sqrt{u^2+1})$ (Logarithm formulas for inverse sinh, inverse cosh, and inverse tanh on their natural domains).

For every real $\alpha$, $u\mapsto u^{\alpha }$ is continuous and differentiable on $(0,\mathrm{\infty })$, with derivative $\alpha u^{\alpha -1}$ (Continuity and derivatives of positive-base real powers).

If $a>0$ and $q\in \mathbb{Q}$, then the real power $a^q$ agrees with the rational power; in particular this holds for $q=1\mathrm{/}2$ (The exponential definition of real powers agrees with the existing rational powers).

Every nonnegative real $a$ has a unique nonnegative square root $\sqrt{a}$ (Square roots exist: a unique $\sqrt{a}\ge 0$ with $(\sqrt{a}{)}^2=a$; the positives are $\{x^2:x\mathrm{\ne }0\}$).

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 $(f\circ g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(g(c))g^{\mathrm{\prime }}(c)$).

Products, sums, and scalar multiples obey their 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 $g(c)\mathrm{\ne }0$).

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).

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 ${\int }_a^{b}f=G(b)-G(a)$).

$\cos 0=1$ and $\cos \pi =-1$ (Quarter-turn values and shifts by pi/2 and pi).