# How to find the area locked between two curves and a line

I need to find the area which is locked between these two curves and a line(the darker area). I have all three equation both curves and the line.

I want just want to composite the right integral for this operation. I will evaluate the area.

• Are you sure this Is about the software Wolfram Mathematica and not mathematics in general? Sep 9, 2022 at 12:43
• Please post all of the functions by Mathematica code at first. Sep 9, 2022 at 12:51

f1[x_] := 1/2*x + 3/2;
f2[x_] := 16/25*x^2 - 16/5*x + 4;
f3[x_] := 6/25*x^2 - 6/5*x + 3/2;

xp1 = x /. First@NSolve[f2[x] == f1[x] && 0 < x < 2.5, x]
yp1 = f1[xp1]

xp2 = 0;
yp2 = f3[xp2];

xp3 = 2.5;
yp3 = 0;

p1=Plot[{f1[x], f2[x], f3[x]}, {x, 0, 5},
PlotLegends -> {"f1", "f2", "f3"}, GridLines -> Automatic,
GridLinesStyle -> LightGray,
Epilog -> {Red, PointSize[0.02],
Point[{{xp1, yp1}, {xp2, yp2}, {xp3, yp3}}]}]


r1 = ImplicitRegion[y > f3[x] && y < f1[x], {{x, 0, xp1}, y}]
r2 = ImplicitRegion[y > f3[x] && y < f2[x], {{x, xp1, 2.5}, y}]

Area@RegionUnion[r1, r2]


p2=RegionPlot[RegionUnion[r1,r2],PlotRange->{{0,5},{0,4}}]
Show[p1,p2]


Thanks @Nasser work out all the formulas.

Clear[f1, f2, f3, plot, reg];
f1[x_] = 1/2*x + 3/2;
f2[x_] = 16/25*x^2 - 16/5*x + 4;
f3[x_] = 6/25*x^2 - 6/5*x + 3/2;
plot = Plot[{f1[x], f2[x], f3[x]}, {x, 0, 5},
PlotStyle -> {Red, Green, Blue}];
reg = ImplicitRegion[{0 <= x <= 5/2, f3[x] <= y <= f2[x],
y <= f1[x]}, {x, y}];
reg // Area
Show[plot,
RegionPlot[DiscretizeRegion@reg, PlotStyle -> Orange,
BoundaryStyle -> Thick]]


14225/12288

Or

x0 = SolveValues[{f1[x] == f2[x], 0 <= x <= 5/2}, x][[1]]
(* 25/32 *)
Integrate[f1[x] - f3[x], {x, 0, x0}] +
Integrate[f2[x] - f3[x], {x, x0, 5/2}]


14225/12288

Use ResourceFunction:

f1[x_] = 1/2*x + 3/2;
f2[x_] = 16/25*x^2 - 16/5*x + 4;
f3[x_] = 6/25*x^2 - 6/5*x + 3/2;

x1 = x /. Solve[f1[x] == f2[x], x] // First;
x2 = x /. Solve[f2[x] == f3[x], x] // First;

area1 = ResourceFunction["AreaBetweenCurves"][{f1[x], f3[x]}, {x, 0, x1}];
area2 = ResourceFunction["AreaBetweenCurves"][{f2[x], f3[x]}, {x, x1, x2}];
area = area1 + area2
14225/12288

area // N
1.15763