Vertical line: plotting and filling

Question

I need some help to complete this code in order to make the wanted plot. In the following, the function that I define as $m(x) = 1$ should be replaced (if possible) to a function that plots a vertical line (equation: $y = 1$) in the range $[-1, 2]$. Then I would like to fill the region under the curve $y = x$ but limited at the right by $y = 1$.

Here is the code I wrote so far, I am stuck on the vertical line + filling part.

Clear[f, g, h, m, plot, reg];
f[x_] = Sqrt[x];
g[x_] = -Sqrt[x];
h[x_] = x;
m[x_] = 1;
plot = Plot[{f[x], g[x], h[x], m[x]}, {x, -1.5, 2}, 
PlotStyle -> {{Darker@Cyan, Dashed}, {Darker@Cyan, Dashed}, 
Darker@Green, Darker@Red}, AspectRatio -> Automatic];
reg = RegionPlot[{h[x] <= y <= m[x]}, {x, -2, 3}, {y, -2, 2}, 
PlotStyle -> LightRed, BoundaryStyle -> None];
Show[reg, plot, PlotRange -> {-2, 3}, AxesStyle -> Arrowheads[{0.05}],
Axes -> True, Frame -> False, 
GridLines -> {{{-2, {Thick, AbsoluteDashing[{3, 3}]}}}, None}, 
Epilog -> {{AbsoluteDashing[3, 3], Line[{{1, 1}, {1, 0}}], 
Line[{{1, 1}, {0, 1}}]}}]

And here is the output:

ADD

Here is the image I would like to replicate (it's from a book).

Answer 1

My attempt:

Clear["Global`*"];

p1 = Plot[{Sqrt[x], -Sqrt[x], x}, {x, -1, 2}
  , AspectRatio -> Automatic
  , PlotStyle -> {
    {Dashing[{0.05, 0.03}], Black, Thick}
    , {Dashing[{0.05, 0.03}], Black, Thick}
    }
  , Ticks -> {None, None}
  , AxesLabel -> {Style["x", Bold, Italic, 20]
    , Style["y", Bold, Italic, 20]
    }
  , AxesStyle -> Directive @@ {Black, Arrowheads[0.07]}
  , Epilog -> {
    {Dashing[{0.05, 0.03}], Black, Thick
     , InfiniteLine[{{1, -1}, {1, 1}}]}
    , AbsolutePointSize[12], Point@{1, 1}
    , Text[Style["1", Bold, 20], {1.1, -0.15}]
    , {Black, Thick
     , Arrow[AnglePath[{{1, 1}, 116.565 Degree}, {{0.75, 0 Degree}}]]
     , Text[Style["∇ f", 20, Bold], {0.5, 1.8}]
     }
    }
  ];

reg = ImplicitRegion[{y <= x && x <= 1 && y < 1}, {x, y}];

p2 = RegionPlot[reg, PlotStyle -> Opacity[0.4, Gray], 
   BoundaryStyle -> {Thick, Black}
   , PlotRange -> {{-2, 2}, {-2, 2}}
   ];

Show[
 p1, p2
 ]

---

Answer 2

$Version

(* "14.0.0 for Mac OS X ARM (64-bit) (December 13, 2023)" *)

Clear["Global`*"]

Using Filling

f[x_] = Sqrt[x];
h[x_] = x;
Plot[{f[x], -f[x], h[x],
  ConditionalExpression[h[x], -1 < x < 1]},
 {x, -1.5, 2},
 AspectRatio -> Automatic,
 PlotStyle -> {
   {Darker@Cyan, Dashed},
   {Darker@Cyan, Dashed},
   {Darker@Green, Dashed},
   Darker@Green},
 Filling -> {4 -> -1},
 FillingStyle -> {None, LightRed},
 PlotRange -> {-1.5, 1.5},
 AxesStyle -> Arrowheads[{0.05}],
 GridLines -> {{{-2, {Thick, AbsoluteDashing[{3, 3}]}}}, None},
 Epilog -> {
   {AbsoluteThickness[1.75], Line[{{1, 1}, {1, -1}}]},
   AbsoluteDashing[3, 3],
   Line[{{1, 2}, {1, -1}}],
   Line[{{1, 1}, {0, 1}}]}]

Answer 3

• Draw a Triangle.

Clear["Global`*"];
f[x_] = Sqrt[x];
g[x_] = -Sqrt[x];
p = {1, f[1]};
q = {1, g[1]};
k = {g[1], g[1]};
plot = Plot[{f[x], g[x]}, {x, -1.5, 2}, 
  PlotStyle -> Directive@{Darker@Cyan, Dashed}, 
  AspectRatio -> Automatic, 
  Epilog -> {AbsoluteThickness[2], Line[{p, q}], Line[{p, k}], Dashed,
     HalfLine[{1, 1}, {0, 1}], HalfLine[{1, 1}, {1, 1}], Gray, 
    Opacity[.5], Triangle[{p, q, k}]}]