(874, #43) Evaluate the quantities in Exercise 39 exactly. The volume of the solid in the first octant bounded by the cylinder and the planes y=z, , and ; .

Solution: The following answers were derived using Maple

a)

mass =

int(int(int(1+x+y+z,z=0..y),y=0..sqrt(1-x^2)),x=0..1);

b) center of mass

First we define , we label it a:

with(plots):

1/(11/24+3/32*Pi):

a:=simplify("):

a*int(int(int(x*(1+x+y+z),z=0..y),y=0..sqrt(1-x^2)),x=0..1);

a*int(int(int(y*(1+x+y+z),z=0..y),y=0..sqrt(1-x^2)),x=0..1):

simplify(");

a*int(int(int(z*(1+x+y+z),z=0..y),y=0..sqrt(1-x^2)),x=0..1):

simplify(");

c) Moment of Inertia

int(int(int((x^2+y^2)*(1+x+y+z),z=0..y),y=0..sqrt(1-x^2)),x=0..1);

by: gm