Triple Integrals

Triple Integrals Triple integrals over a box: By Fubini’s Theorem we can change the order of integration in six possible ways. Applications: 1. 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.)

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.

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:

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:

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.

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

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:

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.

Triple Integrals - Example 3 continued b. 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

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