Mrthdrb

I want to draw the following graphic using tikz:

Is it possible to draw this without using any packages? I can draw the coordinate system as well as both of the arcs but how do I get the space in between filled and also the connections between the arcs give me trouble.

You must load tikz package if you want to use TikZ :)) So I assume that you don't allow to load any additional packages.

In the following proposals, it seems that I don't use any packages. However, I do load one. documentclass[tikz]standalone already load tikz.

The angles are drawn with quotes and angles libraries. They are not packages, so I hope this answer is valid.

documentclass[tikz]standalone
usetikzlibraryquotes,angles
begindocument
begintikzpicture
draw[fill=black!20] (140:1) coordinate (beta) arc (140:-70:1) coordinate (alpha) -- (-70:2) arc (-70:140:2) -- cycle;
draw (-2.5,0)--(2.5,0) coordinate (x);
draw (0,-2.5)--(0,2.5);
fill (1,0) circle (1pt) node[below right] $c^a$ (2,0) circle (1pt) node[below right] $c^b$;
coordinate (o) at (0,0);
pic[draw,<-,"$alpha$",angle radius=0.4cm,angle eccentricity=1.4] angle=alpha--o--x;
pic[draw,->,"$beta$",angle radius=0.6cm,angle eccentricity=1.3] angle=x--o--beta;
endtikzpicture
enddocument

However, if you don't want to load a single library, you can play with arc and node[midway], although this is a bit more difficult

documentclass[tikz]standalone
begindocument
begintikzpicture
draw[fill=black!20] (140:1) coordinate (beta) arc (140:-70:1) coordinate (alpha) -- (-70:2) arc (-70:140:2) -- cycle;
draw (-2.5,0)--(2.5,0) coordinate (x);
draw (0,-2.5)--(0,2.5);
fill (1,0) circle (1pt) node[below right] $c^a$ (2,0) circle (1pt) node[below right] $c^b$;
coordinate (o) at (0,0);
draw[->] (0.4,0) arc (0:-70:0.4) node[midway,below right,inner sep=1pt] $alpha$;
draw[->] (0.6,0) arc (0:140:0.6) node[below right,inner sep=0pt] $beta$;
endtikzpicture
enddocument

Recommended approach:

documentclass[tikz]standalone
usetikzlibraryquotes,angles
begindocument
begintikzpicture
draw[fill=black!20] (140:1) coordinate (beta) arc (140:-70:1) coordinate (alpha) -- (-70:2) arc (-70:140:2) -- cycle;
draw (-2.5,0)--(2.5,0) coordinate (x);
draw (0,-2.5)--(0,2.5);
fill (1,0) circle (1pt) node[below right] $c^a$ (2,0) circle (1pt) node[below right] $c^b$;
coordinate (o) at (0,0);
pic[draw,<-,"$alpha$",angle radius=0.4cm,angle eccentricity=1.4] angle=alpha--o--x;
pic[draw,->,"$beta$",angle radius=0.6cm,angle eccentricity=1.3] angle=x--o--beta;
draw[very thin,dashed] (alpha)--(o)--(beta);
endtikzpicture
enddocument

You can do it with filldraw with some arcs. The use of polar coordinatesd like (-70:1) means starting poiunt is at -70 degrees and distance 1 from origin. The cycleat the end of the path closes the draw command.

documentclass[border=5mm]standalone
usepackagetikz
begindocument
begintikzpicture
 draw (-3,0) -- (3,0);
 draw (0,-3) -- (0,3);
 filldraw[fill=gray!30,opacity=0.7] (-70:1) arc (-70:150:1) -- (150:2) arc (150:-70:2) -- cycle;
 draw[->] (0:0.7) arc (0:-70:0.7)node[pos=0.5,anchor=-35]$alpha$;
 draw[->] (0:0.8) arc (0:150:0.8)node[pos=0.9,anchor=150]$beta$;
endtikzpicture
enddocument