17) Multiple Integrals

a) Double Integrals

e.g. Calculate Int([4-x^2-y^2]-[x^2+y^2]) over the region bounded by x^2+y^2=2.

This is the volume bounded by the paraboloids z=4-x^2-y^2 and z=x^2+y^2 inside the cylinderx^2+y^2=2.

> restart;

> with(plots):

In polar coordinates

> plot3d({[r*cos(t),r*sin(t),4-r^2],[r*cos(t),r*sin(t),r^2]},r=0..sqrt(2),t=0..2*Pi,orientation=[45,65],style=patch);

> Int(Int([4-x^2-y^2]-[x^2+y^2],y=-sqrt(2-x^2)..sqrt(2-x^2)),x=-sqrt(2)..sqrt(2));

> Int(Int(((4-r^2)-(r^2))*r,r=0..sqrt(2)),theta=0..2*Pi);

> value(%);

b) Triple Integrals in Spherical Coordinates

> restart;

> with(mvcal2):

Find the volume of the solid bounded by the sphere x^2+y^2+z^2=2z and an inverted cone with its tip at the origin and making an angle of Pi/6 with the z axis.

> drhodtdphiplot(rho=0..2*cos(phi),theta=0..2*Pi,phi=0..Pi/6,orientation=[45,75],style=patch);

> Int(Int(Int(rho^2*sin(phi),rho=0..2*cos(phi)),theta=0..2*Pi),phi=0..Pi/6);

> value(%);