Τετάρτη, 26 Απριλίου 2017

Proofs without words: Integral of Sine Squared

[asy] import graph; size(220); defaultpen(linewidth(0.7)); Label k; k.p=fontsize(10);  real xmax = pi/2+0.5, xmin = -0.5, ymax = 1.39, ymin = -0.39, lblpt = pi/4 + 0.08;   /* f(x) = sin^2(x) */ real f(real x) { return sin(x) * sin(x); } string pilabel(real x) { if(x > 1) return "$\pi/2$"; else if(x > 0) return "$\pi/4$"; else return "";}  xaxis(xmin,xmax,Ticks(k, pilabel, pi/4),Arrows(6)); yaxis(ymin,ymax,Ticks(k, NoZero),Arrows(6)); filldraw(graph(f,0,pi/2)--(pi/2,0)--(0,0)--cycle,gray(0.7),linewidth(1)); draw((lblpt,f(lblpt))--(lblpt,1),Arrows(6)); draw((lblpt,f(lblpt))--(lblpt,0),Arrows(6)); label("$\cos^2(x)$",(lblpt,f(lblpt)/2+1/2),W,fontsize(10)); label("$\sin^2(x)$",(lblpt,f(lblpt)/2),E,fontsize(10)); draw((0,1)--(pi/2,1),linewidth(1)); [/asy]
$\int_0^{\pi/2} \sin^2 x \, dx = \int_0^{\pi/2} \cos^2 x \, dx = \frac {\pi}{4}$ 
from
 $\begin{cases}\sin^2 x + \cos^2 x = 1\\ \sin x = \cos(\pi/2 - x)\end{cases}$.

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