1. Nájdite dvojný integrál funkcie f(x,y)=3x^2*y na oblasti M=[1,3]x[2,5].

Integrate[3x^2*y,{x,1,3}]

Integrate[26*y,{y,2,5}]

Integrate[3x^2*y,{y,2,5}]

Integrate[63x^2/2,{x,1,3}]

Integrate[3x^2,{x,1,3}]*Integrate[y,{y,2,5}]

Plot3D[3  x^2*y, {x, 1, 3}, {y, 2, 5}]

2. Nájdite dvojný integrál funkcie f(x,y)=(2*x^3+5)*Sin[y] na oblasti M=[0,1]x[0,Pi].

f[x_,y_]=(2*x^3+5)*Sin[y]
Integrate[f[x,y],{x,0,1},{y,0,Pi}]
Integrate[f[x,y],{y,0,Pi},{x,0,1}]
Plot3D[f[x,y],{x,0,1},{y,0,Pi}]

3. Nájdite dvojný integrál funkcie f(x,y)=2/Sqrt[4-y^2] na oblasti M={[x,y],0<=x<=Sqrt[4-y^2],0<=y<=2}.

Integrate[2/Sqrt[4-y^2],{x,0,Sqrt[4-y^2]},{y,0,2}]
Integrate[2/Sqrt[4-y^2],{y,0,2},{x,0,Sqrt[4-y^2]}]
Plot3D[2/Sqrt[4-y^2],{x,0,2},{y,0,2}]

4. Nájdite dvojný integrál funkcie f(x,y)=Log[x+y] na oblasti M=[y,2*y]x[1,2].

Integrate[Log[x+y],{y,1,2},{x,y,y*2}]
Plot3D[Log[x+y],{x,1,4},{y,0,2}]

5. Nájdite dvojný integrál funkcie f(x,y)=Exp[-x-y] na oblasti A určenej nerovnosťami x>=0,y>=0,x+y<=1.

Plot[{1-x},{x,0,1}]

Integrate[Exp[-x-y],{x,0,1},{y,0,1-x}]

Plot3D[Exp[-x-y],{x,0,1},{y,0,1}]

ParametricPlot3D[{u,v,Exp[-u-v]},{u,0,1},{v,0,1}]
ParametricPlot3D[{1-u,u,v},{u,0,1},{v,0,1}]
Show[%,%%]

6. Nájdite trojný integrál funkcie f(x,y,z)=x^2*y^2*z na oblasti M=[0,3]x[0,2]x[2,5].

g[x_,y_,z_]=x^2*y^2*z

Plot3D[2,{x,0,3},{y,0,2}]
Plot3D[5,{x,0,3},{y,0,2}]
Show[%,%%]

Integrate[x^2*y^2*z,{x,0,3},{y,0,2},{z,2,5}]
Integrate[x^2,{x,0,3}]*Integrate[y^2,{y,0,2}]*Integrate[z,{z,2,5}]

7. Nájdite trojný integrál funkcie f(x,y,z)=Cos[x+y+z] na oblasti M=[0,Pi]x[0,Pi]x[0,Pi/2].

Integrate[Cos[x+y+z],{x,0,Pi}]

Integrate[-2*Sin[y+z],{y,0,Pi}]

Integrate[-4*Cos[z],{z,0,Pi/2}]

Integrate[Cos[x+y+z],{x,0,Pi},{y,0,Pi},{z,0,Pi/2}]

8. Nájdite trojný integrál funkcie f(x,y,z)=z na oblasti M={[x,y,z],0<=x<=1,0<=y<=3-3*x,0<=z<=3-3*x-y}.

Integrate[z,{x,0,1},{y,0,3-3*x},{z,0,3-3*x-y}]
Plot[3-3*x,{x,0,1}]
Plot3D[3-3*x-y,{x,0,1},{y,0,3}]

ParametricPlot3D[{u, 3 - 3*v, 0}, {u, 0, 1}, {v, 0, 1}]
ParametricPlot3D[{u, 3 - 3*v, 3 - 3*u - 3*v}, {u, 0, 1}, {v, 0, 
        1}]
Show[%, %%, ViewPoint -> {-3.661,  2.003,  1.363}]

9.  Nájdite trojný integrál funkcie f (x, y, z) = x*y*
          z na oblasti M = {[x, y, z], 
            1 ≤ x ≤ z, 
            0 ≤ y ≤ 2, -1 ≤ 
              z ≤ y^2}.

Integrate[x*y*z, {y, 0, 2}, {z, -1, y^2}, {x, 1, z}]

Plot3D[-1, {x, 1, 4}, {y, 0, 2}]
Plot3D[x, {x, 1, 4}, {y, 0, 2}]
Plot3D[y*y, {x, 1, 4}, {y, 0, 2}]
Show[%, %%, %%%, 
      ViewPoint -> {-3.335,  1.562,  2.390}]

10. Nájdite trojný integrál funkcie f(x,y,z)=x+y+z na oblasti M={[x,y,z],-Sqrt[1-y^2-z^2]<=x<=Sqrt[1-y^2-z^2],-Sqrt[1-z^2]<=y<=Sqrt[1-z^2],-1<=z<=1}.

Integrate[x+y+z,{z,-1,1},{y,-Sqrt[1-z^2],Sqrt[1-z^2]},{x,-Sqrt[1-y^2-z^2],Sqrt[1-y^2-z^2]-1,y^2}]

Plot3D[Sqrt[1 - x^2 - y^2], {x, 0, 1}, {y, -1, 1}]