Ellipse - Weyl's law - Mathematica

I don't have experience with Mathematica. I'd need that someone could help me. Could anyone be able to reproduce the following image by this software? Here, the blue quarter ellipse is completely outside the bold region, while the red one is inside. sol = Solve[{
x^2/a^2 + y^2/b^2 == 1 /. {x -> 5, y -> 2},
x^2/a^2 + y^2/b^2 == 1 /. {x -> 4, y -> 3},
a > 0, b > 0}, {a, b}][]

(*  {a -> Sqrt[161/5], b -> Sqrt/3}  *)

x^2/a^2 + y^2/b^2 == 1 /. sol

(*  (5*x^2)/161 + (9*y^2)/161 == 1  *)

Graphics[{
Gray,
Opacity[.25],
Rectangle[{#, 0}] & /@ Range[0, 4],
Rectangle[{#, 1}] & /@ Range[0, 4],
Rectangle[{#, 2}] & /@ Range[0, 3],
Opacity, Thick,
Red,
Tooltip[
Circle[{0, 0}, {5, 3}, {0, Pi/2}],
x^2/a^2 + y^2/b^2 == 1 /. {a -> 5, b -> 3}],
Blue,
Tooltip[
Circle[{0, 0}, {a, b} /. sol, {0, Pi/2}],
x^2/a^2 + y^2/b^2 == 1 /. sol]},
GridLines -> {Range[0, 7], Range[0, 5]},
Axes -> True,
AxesLabel -> (Style[#, 14, Bold] & /@ {"x", "y"})] • The bold part is too light. Would you be able to blacken a little? – CHSLD Aug 8 '16 at 21:31
• @CHSLD - I can't guess what color/darkness you want. Change Opacity[.25] to a larger number[e.g., .6) and/or change the color from LightGrayto Gray or any other color the you want. – Bob Hanlon Aug 8 '16 at 21:46
• Just the normal Gray, not a LightGray. I don't have Mathematica. I just need the image – CHSLD Aug 8 '16 at 21:49
• @CHSLD "I don't have Mathematica. I just need the image" - well next time, you can do it yourself. – J. M. will be back soon Aug 9 '16 at 4:45
Show[ContourPlot[x^2/6^2 + y^2/4^2 == 1, {x, 0, 6}, {y, 0, 4}, GridLines -> Automatic],
ContourPlot[x^2/5^2 + y^2/3^2 == 1, {x, 0, 5}, {y, 0, 4},  ContourStyle -> Red],
Table[Graphics[{Opacity[0.15], Rectangle[{x, y}]}], {x, 0, 3}, {y, 0,
2}], Graphics[{Opacity[0.15], Rectangle[{4, 0}]}], Graphics[{Opacity[0.15], Rectangle[{4, 1}]}]] • Could you insert the tiles and the bold part as the esquisse? – CHSLD Aug 8 '16 at 20:01
• Sure thing man! – M Horsley Aug 8 '16 at 20:20