Trjtdtk

MWE:

documentclass[tikz,border=10pt]standalone
usepackagetikz
usetikzlibrarydatavisualization
usetikzlibraryarrows,calc
usetikzlibraryshapes,positioning
usetikzlibrarydecorations.markings
usetikzlibrarydatavisualization.formats.functions
defmytypesetter#1
pgfmathparse#1/pi
pgfkeys/pgf/number format/precision=2
pgfmathroundtozerofillpgfmathresult
pgfmathifisintpgfmathresult
pgfmathprintnumberpgfmathresult$pi$

pgfmathprintnumber[/pgf/number format/frac, frac denom=2, frac whole=false]pgfmathresult$pi$


begindocument


begintikzpicture[>=latex]
datavisualization [
 school book axes, %scientific axes=clean,
 all axes=,
 x axis=
 label=$omega t$,
 ticks=
 step=(pi/2),
 tick typesetter/.code=mytypesetter##1,
 ,
 y axis=
 label=$v$, 
 ticks=none,
 ,
 style sheet=strong colors,
 style sheet=vary dashing,
 visualize as smooth line/.list=voltage,
 voltage=label in legend=text=$v$,
]
data [set=voltage, format=function] 
 var x : interval [0:2*pi] samples 100;
 func y = 2*sin(value x r);
;
endtikzpicture 
enddocument

But this is what I want

.

How do I get this graph from my source code without using axis environment?

Nothing prevents you from adding the fills on the background.

documentclass[tikz,border=10pt]standalone
usetikzlibrarydatavisualization
usetikzlibrarydatavisualization.formats.functions
usetikzlibrarypatterns,backgrounds
defmytypesetter#1
pgfmathparse#1/pi
pgfkeys/pgf/number format/precision=2
pgfmathroundtozerofillpgfmathresult
pgfmathifisintpgfmathresult
pgfmathprintnumberpgfmathresult$pi$

pgfmathprintnumber[/pgf/number format/frac, frac denom=2, frac whole=false]pgfmathresult$pi$


begindocument


begintikzpicture[>=latex]
datavisualization [
 school book axes, %scientific axes=clean,
 all axes=,
 x axis=
 label=$omega t$,
 ticks=
 step=(pi/2),
 tick typesetter/.code=mytypesetter##1,
 ,
 y axis=
 label=$v$, 
 ticks=none,
 ,
 style sheet=strong colors,
 style sheet=vary dashing,
 visualize as smooth line/.list=voltage,
 voltage=label in legend=text=$v$,
]
data [set=voltage, format=function] 
 var x : interval [0:2*pi] samples 100;
 func y = 2*sin(value x r);
;
beginscope[on background layer]
 fill[red] plot[variable=x,domain=0:pi] (x,2*sin(x r));
 path[pattern=horizontal lines,pattern color=red] 
 plot[variable=x,domain=pi:2*pi] (x,2*sin(x r));
endscope 
endtikzpicture 
enddocument

This lets you keep what you had and add the fills where needed.

Let me also comment that Symbol 1's nice answer can be used here. I stress that I have close to zero knowledge on the data visualization business, so I may make things unnecessarily complicated. The strategy is from Symbol 1's nice answer, except that I allow the style to be redefined via tikzsetchange style=draw,pattern=horizontal lines,pattern color=red (yes, I know that is a stupid name for a pgf key, yet I feel that this is not the only thing that I am doing stupid here. This works, but is far more complicated than JouleV's nice code. There is a good chance that (at least) part of the complexity is due to my ignorance with regards to data visualization.

documentclass[tikz,border=10pt]standalone
usepackagetikz
usetikzlibrarydatavisualization
usetikzlibrarypatterns
usetikzlibrarydatavisualization.formats.functions
defmytypesetter#1
pgfmathparse#1/pi
pgfkeys/pgf/number format/precision=2
pgfmathroundtozerofillpgfmathresult
pgfmathifisintpgfmathresult
pgfmathprintnumberpgfmathresult$pi$

pgfmathprintnumber[/pgf/number format/frac, frac denom=2, frac whole=false]pgfmathresult$pi$


makeatletter%from https://tex.stackexchange.com/a/228265/121799
tikzdatavisualizationset
 visualize as generic/.style=
 new object=
 when=after survey,
 store=/tikz/data visualization/visualizers/#1,
 class=plot handler visualizer,
 arg1=#1,
 arg2=tikz@dv@plot@handler,tikz@dv@plot@mark@maker
 ,
 new visualizer=#1%
 every path/.style=generic style,
 style=every mark/.append style=color=visualizer color,
 mark size=2pt,
 semithick,
 color=visualizer color,
 mark=none,
 /tikz/data visualization/every visualize as line/.try,
 visualizer in legend=tikz@dv@legend@entry@as@example,
 #1=straight line
 ,
 visualize as generic/.default=generic,

tikzsetchange style/.code=tikzsetgeneric style/.style=#1,
change style=draw,fill=red
makeatother
begindocument


begintikzpicture[>=latex]
datavisualization [
 school book axes, %scientific axes=clean,
 all axes=,
 x axis=include value=6.3,
 label=$omega t$,
 ticks=
 step=(pi/2),
 tick typesetter/.code=mytypesetter##1,
 ,
 y axis=include value=-2,
 label=$v$, 
 ticks=none,
 ,
 style sheet=strong colors,
 style sheet=vary dashing,
 visualize as generic,
 %visualize as smooth line/.list=voltage,
% voltage=label in legend=text=$v$,
]
data [format=function] % set=voltage,
 var x : interval [0:pi] samples 100;
 func y = 2*sin(value x r);
 ;
tikzsetchange style=draw,pattern=horizontal lines,pattern color=red
datavisualization [
 school book axes, %scientific axes=clean,
 all axes=,
 x axis=include value=6.3,
 label=$omega t$,
 ticks=
 step=(pi/2),
 tick typesetter/.code=mytypesetter##1,
 ,
 y axis=include value=2,
 label=$v$, 
 ticks=none,
 ,
 style sheet=strong colors,
 style sheet=vary dashing,
 visualize as generic,
 %visualize as smooth line/.list=voltage,
% voltage=label in legend=text=$v$,
]
data [format=function]
 var x : interval [pi:2*pi] samples 100;
 func y = 2*sin(value x r);
;
endtikzpicture 
enddocument

A pure TikZ solution for fun, which is much shorter. I have no idea why you have to use datavisualization for this.

documentclass[tikz]standalone
usetikzlibrarypatterns
begindocument
begintikzpicture[>=latex,y=2cm]
draw[smooth,fill=red] plot[samples=100,domain=0:pi] (x,sin(deg(x)));
draw[smooth,pattern=horizontal lines,pattern color=red] plot[samples=100,domain=pi:2*pi] (x,sin(deg(x)));
draw[->] (0,-1.1)--(0,1.1) node[above] $v$;
draw[->] (-.2,0)--(2*pi+0.5,0) node[right] $omega t$;
draw (pi/2,-.05) node[below] $frac12pi$--(pi/2,.05);
draw (pi,-.05) node[below] $1pi$--(pi,.05);
draw (3*pi/2,-.05) node[below] $frac32pi$--(3*pi/2,.05);
draw (2*pi,-.05) node[below] $2pi$--(2*pi,.05);
draw (8.5,0) node tikzdraw[x=1pt,y=1pt] (0,0) to[out=30,in=-150] (20,0); $v$;
endtikzpicture
enddocument