Quiz (Triple Integrals)

Show all work and justify each step.

(1) Find the volume of the solids bounded by the graphs of the equations by using triple integration.
    
    (a) The solid bounded by the sphere and the paraboloid
    
    (b) The solid of the region bounded by the cylinders and and the planes and
    
    (c) Find the volume of the region between the two elliptic paraboloids and

(2) Change the order of integration to show that



Also, show that

(3) Higher-dimensional multiple integrals can be defined and evaluate in essentially the same way as double integrals and triple integrals. Evaluate the following multiple integrals.

    (a) where is the four-dimensional "hyperbox" defined by and
    
    (b) where is the four-dimensional region bounded by the hyperplane and the coordinate spaces and in the first hyperoctant (where ).
    

• print this page
• pages linking here
• related pages