1. Triple Integrals

2. Triple Integrals Triple integrals over a box:
By Fubini’s Theorem we can change the order of integration in six possible ways.
Applications:
is the volume of the solid E.
If 𝑓(𝑥,𝑦,𝑧) is the mass density of the solid E (e.g. in kg/m3), then
is the total mass of the solid (in kg.)

3. Triple integrals Example: Evaluate where E is the solid defined by
By Fubini’s Theorem we can change the order of integration:
All these integrals give the same result.

4. Triple Integrals Type 1 Triple integrals over more general regions:
There are six different orders of integration possible in a triple iterated integral.
Type 1: Let D be the projection of the solid on the xy-plane and let 𝑧= 𝑢 1 (𝑥,𝑦) and 𝑧= 𝑢 2 (𝑥,𝑦) be the surfaces forming the “bottom” and the “top” of the solid respectively.
There are two different orders of integration on D. For instance, as a type I region we obtain the integral:

5. Triple Integrals Type 2
Let D be the projection of the solid on the yz-plane and let 𝑥= 𝑢 1 (𝑦,𝑧) and
𝑥= 𝑢 2 (𝑦,𝑧) be the surfaces forming the “back” and “front” of the solid.
There are two different orders of integration on D. For instance, as a type I region we obtain the integral:

6. Triple Integrals Type 3
Let D be the projection of the solid on the xz-plane and let 𝑦= 𝑢 1 (𝑥,𝑧) and
𝑦= 𝑢 2 (𝑥,𝑧) be the surfaces forming the “left” and “right” sides of the solid.
There are two different orders of integration on D. For instance, as a type I region we obtain the integral:
Remarks:
The limits of integration for the middle integral can involve only the outmost variable of integration.
The outside limits must be constant.

7. Triple Integrals Example 1
Evaluate where E is the tetrahedron bounded by the coordinate planes and
The solid is bounded below by 𝑧=0 and above by
Let D be the projection of the tetrahedron on the xy-plane.
The arrow enters and exits the solid at the limit of integration for z D

8. Triple Integrals Example 2
Use a triple integral to find the volume of the solid bounded by the paraboloid 𝑥=4 𝑦 2 +4 𝑧 2 and the plane x = 4.
A line parallel to the x-axis intersects the solid at 𝑥=4 𝑦 2 +4 𝑧 2 and at x = 4. These are the limits of integration for x.
Let D be the projection of the solid in the yz-plane.
𝒙 𝟐 + 𝒛 𝟐 =𝟏
The surfaces 𝑥=4 𝑦 2 +4 𝑧 2 and x = 4 intersect on a curve C: 4y2 + 4z2 = 4 → y2 + z2 = 1
The circle is the boundary of the region D:
polar coordinates:

9. Triple Integrals - Example 3
Let E be the solid bounded by 𝑧=0, 𝑧=𝑦 and 𝑦=9− 𝑥 2 .
Express in the form: a.
Limits for z: 0≤𝑧≤𝑦
Let D be the projection of the solid on the 𝑥𝑦-plane.

10. Triple Integrals - Example 3 continuedb.
A line parallel to the y-axis intersects the solid on the surface 𝑦=𝑧 (left surface) and on the surface
𝑦=9− 𝑥 2 (right surface). These are the limits of integration for the y-variable.
Let D be the projection of the solid on the 𝑥𝑧-plane.
The arrow enters and exits the solid at the limit of integration for y
The surfaces z = y and y = 9 − x2 intersect in a curve C. The projection of this curve on the xz plane has equation z = 9 – x2 and it is the boundary of the domain D

11. Triple Integrals - Example 3 continued
A line parallel to the x-axis intersects the solid on the surface 𝑥=− 9−𝑦 (back surface) and on the surface 𝑥= 9−𝑦 (front surface). These are the limits of integration for the x-variable.
Let D be the projection of the solid on the 𝑦𝑧-plane. y
The arrow enters and exits the solid at the limit of integration for x