Integration examples

ClearAll["Global`*"];

Integrate to find the blue area: 
This second (shaded) picture is with σ = x y:

y1[x_]=2+0.5 x;y2[x_]=1.5+3 x - (x^2);
Plot[{y1[x],y2[x]},{x,0,3}]
x1=x/.FindRoot[y1[x]==y2[x],{x,0.0}][[1]];
x2=x/.FindRoot[y1[x]==y2[x],{x,2.0}][[1]];

Print["The horizontal intersection values are (Subscript[x, 1]) = ",x1," and (Subscript[x, 2]) = ",x2]
A1 = (∫_x1^x2)(∫_(y1[x])^(y2[x]))1\[DifferentialD]y\[DifferentialD]x ;
Print ["The area of the blue shape is ",A1] 
xcm=(((∫_x1^x2)(∫_(y1[x])^(y2[x]))x\[DifferentialD]y\[DifferentialD]x)/A1);
Print["The horizontal center of area of the uniform shape is ",xcm]
ycm=(((∫_x1^x2)(∫_(y1[x])^(y2[x]))y\[DifferentialD]y\[DifferentialD]x)/A1);
Print["The vertical center of area of the uniform shape is ",ycm]
σ=x y; (* i.e., density is not constant *)
m1 = (∫_x1^x2)(∫_(y1[x])^(y2[x]))σ\[DifferentialD]y\[DifferentialD]x ;
Print["The total mass of the weighted shape is ",m1]
xcmw=(((∫_x1^x2)(∫_(y1[x])^(y2[x]))σ x\[DifferentialD]y\[DifferentialD]x)/((∫_x1^x2)(∫_(y1[x])^(y2[x]))σ \[DifferentialD]y\[DifferentialD]x));
Print["The horizontal center of mass of the weighted shape is ",xcmw]
ycmw=(((∫_x1^x2)(∫_(y1[x])^(y2[x]))σ y\[DifferentialD]y\[DifferentialD]x)/((∫_x1^x2)(∫_(y1[x])^(y2[x]))σ \[DifferentialD]y\[DifferentialD]x));
Print["The vertical center of mass of the weighted shape is ",ycmw]

ClearAll["Global`*"];

Integrate to find the volume:
For this problem, the limits are the full rectangular field: -1 < x < +1, and -1 < y < +1.

z2[x_,y_]=(1-(x^2))(1-(y^2));
V2= 4.0(∫_0^1)(∫_0^1)z2[x,y]\[DifferentialD]y\[DifferentialD]x ; (* uses symmetry of the 4 quadrants...*)
Print["The volume of this bump is ",V2]

ClearAll["Global`*"];

Integrate to find the volume:
For this problem, the limits of integration are only within the circular region described by (x^2)+(y^2)<1.

z3[x_,y_]=(1-(x^2))(1-(y^2)); (* this is the same function as the previous one ! *)
V3= 4.0(∫_0^1)(∫_0^(Sqrt[1-(x^2)]))z3[x,y]\[DifferentialD]y\[DifferentialD]x ; (* again uses symmetry of the 4 quadrants...*)
Print["The volume of this bump is ",V3]