(874, #44) Evaluate the quantities in Exercise 42 to three decimal places. The volume of the solid in the first octant bounded by the cylinder and the plane , , and in the first octant; .

Solution: The following answers we derived using Maple. The codes are provided following the iterated integral.

mass =

evalf(int(int(int(x^2+y^2,z=0..sqrt(9-y^2)),y=3*x..3),x=0..1),3);

11.2

To find the center of mass we must first find the actual mass. The following is the code used by maple to find the exact mass.

Center of mass

int(int(int(x^2+y^2,z=0..sqrt(9-y^2)),y=3*x..3),x=0..1);

56/5

evalf((5/56)*int(int(int(x*(x^2+y^2),z=0..sqrt(9-y^2)),y=3*x..3),x=0..1),3);

.374

evalf((5/56)*int(int(int(y*(x^2+y^2),z=0..sqrt(9-y^2)),y=3*x..3),x=0..1),3);

2.21

evalf((5/56)*int(int(int(z*(x^2+y^2),z=0..sqrt(9-y^2)),y=3*x..3),x=0..1),3);

.938

Moment of inertia

evalf(int(int(int((x^2+y^2)*(x^2+y^2),z=0..sqrt(9-y^2)),y=3*x..3),x=0..1),3);

59.8

by: gm