This command generates a picture of the area between f(x) = 2-(x^2) and g(x) = x:

p1 = Plot[{2-x^2, x}, {x, -3, 3}, PlotStyle -> {{Thick, Red}, {Thick, Blue}}, Filling -> {2 -> {1}}, FillingStyle -> {Yellow, White}, AspectRatio -> Automatic, PlotRange -> {-3, 3}]

This command generates an animation illustrating the idea of calculating the above area by adding an infinite number of rectangles of infinitesimally small width:

Manipulate[ParametricPlot[{{d, (2-d^2)(t+3)(1/6) + (1/6)(3-t)d},{t,t},{t,2 - t^2}},{t,-3,3}, PlotRange ->{{-3,3},{-3,3}} , PlotStyle -> {Directive[Green, Thick], Directive[Red, Thick], Directive[Blue, Thick]}], {d,-2,1}]

f[x_] = x^3 + x^2 - 6x;
g[x_] = -x^3 + 5x^2;
Plot[{f[x], g[x]}, {x, -4, 4}, PlotStyle -> {{Red, Thick}, {Blue, Thick}}, PlotRange -> {-10, 30}]

Solve[f[x] == g[x], x]

r1[x_] =Which[x > 0 && x < 2, 2x];
r2[x_] = Which[x > 2 && x <3, 4];
r3[x_] = Which[x > 1 && x < 3, (x-1)^2 ];
r4[x_] =Which[x > 0 && x < 1, 0];

Plot[{r1[x], r2[x], r3[x], r4[x]}, {x, -1, 4}, PlotRange -> {-1, 4}, AspectRatio -> Automatic, PlotStyle -> {{Thick, Purple}, {Thick, Darker[Green]}, {Thick, Blue}, {Thick,Red}}, AxesLabel -> {x,y}]

Plot[{r1[x], r2[x], r3[x], r4[x]}, {x, -1, 4}, PlotRange -> {-1, 4}, AspectRatio -> Automatic, PlotStyle -> {{Thick, Purple}, {Thick, Darker[Green]}, {Thick, Blue}, {Thick,Red}}, AxesLabel -> {x,y}, Filling -> {4 -> {1}, 1 -> {3}, 3 -> {2}}, FillingStyle -> {Yellow,Lighter[ Orange]}]

This shows pictorially the method of x-integration for the above region:

Manipulate[ParametricPlot[{{d ,Which[d < 1, 2 d t, d < 2, (d-1)^2 t + 2d (1-t), d < 3, (d-1)^2 t +  4(1-t) ]},{2t,4t},{t+2,4},{t,0},{2t+1,4t^2}},{t,0,1}, PlotRange ->{{-1,4},{-1,5}} , PlotStyle -> {Directive[Black, Thick], Directive[Purple, Thick], Directive[Darker[Green], Thick], Directive[Red, Thick], Directive[Blue, Thick]}], {d,0,3}]

This shows pictorially the method of y-integration for the above region:

Manipulate[ParametricPlot[{{(1/2) d t + (1-t)( Sqrt[d] + 1), d},{2t,4t},{t+2,4},{t,0},{2t+1,4t^2}},{t,0,1}, PlotRange ->{{-1,4},{-1,5}} , PlotStyle -> {Directive[Black, Thick], Directive[Purple, Thick], Directive[Darker[Green], Thick], Directive[Red, Thick], Directive[Blue, Thick]}], {d,0,4}]

This shows the x-integration and y-integration on the same picture:

Manipulate[{ParametricPlot[{{d ,Which[d < 1, 2 d t, d < 2, (d-1)^2 t + 2d (1-t), d < 3, (d-1)^2 t +  4(1-t) ]},{2t,4t},{t+2,4},{t,0},{2t+1,4t^2}},{t,0,1}, PlotRange ->{{-1,4},{-1,5}} , PlotStyle -> {Directive[Black, Thick], Directive[Purple, Thick], Directive[Darker[Green], Thick], Directive[Red, Thick], Directive[Blue, Thick]}], ParametricPlot[{{(1/2) c t + (1-t)( Sqrt[c] + 1), c},{2t,4t},{t+2,4},{t,0},{2t+1,4t^2}},{t,0,1}, PlotRange ->{{-1,4},{-1,5}} , PlotStyle -> {Directive[Black, Thick], Directive[Purple, Thick], Directive[Darker[Green], Thick], Directive[Red, Thick], Directive[Blue, Thick]}]}, {c, 0, 4},  {d,0,3}]

This shows the picture after the x - and y - axes have been transposed :

s1[y_] = Which[y > 0 && y < 4, (1/2) y];
s2[y_] = Which[y > 0 && y < 4, (Sqrt[y]) + 1];

Plot[{s1[y], s2[y]}, {y, -1, 5}, PlotRange -> {-1, 4}, AspectRatio -> Automatic, Filling -> {1 -> {2}}, FillingStyle -> Yellow, PlotStyle -> {{Purple, Thick}, {Blue, Thick}}, AxesLabel -> {y,x}]