To calculate double or multiple integrals one needs to build iterative integrals and then to calculate them.

"Maple" can help you to evaluate simple iterative integrals. You need to use the student-package.

First load the package

>with(student):

Then you can use the following "Maple" procedures

>value(Doubleint(g(x,y),x=a(y)..b(y),y=c..d));

>value(Tripleint(f(x,y,z),x=a(y,z)..b(y,z),y=c(z)..d(z),z=g..h));

in order to evaluate double and triple integrals, respectively. Notice that first you need to set up limits of integration correctly and then use the "Maple" procedures. If, for some reason, you decide to change the order of integration, then you need to recalculate all the limits of integration.

1. An ice cream cone is the region bounded by the hemisphere z=sqrt(16-x^2 - y^2) and the cone z = sqrt(x^2 + y^2). Find its volume.
2. Calculate the volume of the solid defined by the inequalities
   -sqrt(4-x^2-y^2)< z < sqrt(4-x^2-y^2),

   x^2+y^2 < 1

3. Calculate the volume of the solid defined by the inrqualities
   -sqrt(4-x^2-y^2)< z < y^2 +1

   x^2+y^2 < 1

4. Find the volume of the region bounded above by the paraboloid

   z = 5 - x^2 - y^2

   and below by the paraboloid

   z = 4x^2 + 4 y^2

5. Find the volume of the region bounded above by the sphere

   x^2 + y^2 + z^2 =2

   and below by the circular paraboloid

   z = x^2 + y^2