1730.sagews

var('x, y')
f=e^(-x^2);
P=plot(f, [0, 2]);
Q=plot(x, [0, 1.5], linestyle='--', rgbcolor=(1,0,0));
P1=polygon([(0,0),(0,1)]+[(y,f.subs_expr(x==y)) for y in [0,0.1,..,2]]+[(0,0),(2,0)]+[(2,0),(2, f(2))], alpha=0.1);
show(P+Q+P1, aspect_ratio=1, figsize=[5,5], axes_labels=('x','y'));
reset()

reset()

var('x, y')
g=sqrt(-log(x));
R=plot(sqrt(-log(x)), [0, 1]);
S=plot(x, [0, 1.8], linestyle='--', rgbcolor=(1,0,0));
T1=polygon([(0,0),(0,2)]+[(0,2),(e^(-4),2)]+[(y,g.subs_expr(x==y)) for y in [e^(-4),e^(-4)+0.05,..,1]]+[(1,0),(0,0)], alpha=0.1);
show(R+S+T1, aspect_ratio=1, figsize=[5,5], axes_labels=('y','x'));
reset()

var('x,y');
show((2/pi)*((x/sqrt(1+x^2))*arctan(y/sqrt(1+x^2))+(y/sqrt(1+y^2))*arctan(x/sqrt(1+y^2))));

\newcommand{\Bold}[1]{\mathbf{#1}}\frac{2 \, {\left(\frac{y \arctan\left(\frac{x}{\sqrt{y^{2} + 1}}\right)}{\sqrt{y^{2} + 1}} + \frac{x \arctan\left(\frac{y}{\sqrt{x^{2} + 1}}\right)}{\sqrt{x^{2} + 1}}\right)}}{\pi}

var('x,y');
F=(2/pi)*((x/sqrt(1+x^2))*arctan(y/sqrt(1+x^2))+(y/sqrt(1+y^2))*arctan(x/sqrt(1+y^2)));
plot3d(F, (x,-1,1), (y,-1,1));

reset()

var('x');
show((2/pi)*((x/sqrt(1+x^2))*arctan(x/sqrt(1+x^2))+(x/sqrt(1+x^2))*arctan(x/sqrt(1+x^2))));

\newcommand{\Bold}[1]{\mathbf{#1}}\frac{4 \, x \arctan\left(\frac{x}{\sqrt{x^{2} + 1}}\right)}{\sqrt{x^{2} + 1} \pi}