This document provides two examples of using double and triple integrals to calculate the moment of inertia and volume of solids. The first example calculates the moment of inertia of a solid inside a cylinder using cylindrical coordinates. The second example finds the volume of a solid inside a sphere and outside a cone using spherical coordinates. It converts the equations to spherical coordinates and sets up the integral to evaluate the volume.

1. DOUBLE AND TRIPLE
INTEGRAL
Sonendra Gupta
Associate Professor
Department of mathematics
Author of 17 Books of Mathematics
Visit:sonendragupta.blogspot.in

2. INTRODUcTION

15. ExamplE
1.) Find the moment of inertia about the z-axis of the solid that lies below the paraboloid
z = 25 – x2
- y2
inside the cylinder
x2
+ y2
= 4
above the xy-plane, and has density function
ρ(x,y,z) = x2
+ y2
+ 6z
Solution
By the moment of inertia formula, we have
The region, being inside of a cylinder is ripe for cylindrical coordinates. We get

19. Example
Find the volume of solid that lies inside the sphere
x2
+ y2
+ z2
= 2
and outside of the cone
z2
= x2
+ y2
Solution
We convert to spherical coordinates. The sphere becomes
ρ =
To convert the cone, we add z2
to both sides of the equation
2z2
= x2
+ y2
+z2
Now convert to
2ρ2
cos2
φ = ρ2
Canceling the ρ2
and solving for φ we get
φ = cos-1
(1/ ) = π/4 or 7π/4
In spherical coordinates (since the
coordinates are π periodic)
7π/4 = 3π/4

20. To find the volume we compute
3
2 24
2
0 0
4
sinV d d d
π
π
π
ρ φ ρ φ θ= ∫ ∫ ∫
Evaluating this integral should be routine at this point and is equal to
8π
V = —
3