Triple Integrals in Rectangular Coordinates

University of Babylon Lecture: Hussein ALbermany
Collage of Material Engineering Subject: Mathematics
Department of Metallurgy Engineering Stage : 2nd stage
17
Triple Integrals in Rectangular Coordinates
triple integrals enable us to solve still more general problems. We use triple integrals to calculate the volumes of three-dimensional shapes, the masses and moments of solids of varying density, and the average value of a function over a three dimensional region.
Triple Integrals
If F(x, y, z) is a function defined on a closed bounded region D in space.
We call this limit the triple integral of F over D and write
Volume of a Region in Space.
The volume of a closed, bounded region D in space is
Properties of Triple Integrals
University of Babylon Lecture: Hussein ALbermany
Collage of Material Engineering Subject: Mathematics
Department of Metallurgy Engineering Stage : 2nd stage
18
Finding Limits of Integration
1. Sketch: Sketch the region D along with its “shadow” R (vertical projection) in the xy-plane. Label the upper and lower bounding surfaces of D and the upper and lower bounding curves of R.
2. Find the z-limits of integration: Draw a line M passing through a typical point (x, y) in R parallel to the z-axis. As z increases, M enters D at z = ƒ1(x, y) and leaves at z = ƒ2(x, y) These are the z-limits of integration.
3. Find the y-limits of integration: Draw a line L through (x, y)
parallel to the y-axis. As y increases, L enters R ,at y = g1(x) and leaves at y = g2(x) . These are the y-limits of integration.
4. Find the x-limits of integration: Choose x-limits that include all lines through R parallel to the y-axis (x = a and x = b in the preceding figure). These are the x-limits of integration. The integral is
University of Babylon Lecture: Hussein ALbermany
Collage of Material Engineering Subject: Mathematics
Department of Metallurgy Engineering Stage : 2nd stage
19
EXAMPLE 11: Find the volume of the region D enclosed by the surfaces z = x2 + 3y2 and z = 8-x2 - y2.
Solution:
1. We first sketch the region
2. find the z-limits of integration
enters z1 = x2 + 3y2 , leaves z2 = 8-x2 - y2.
3. find the y-limits of integration
x2 + 3y2 = 8-x2 - y2
[ 2x2 + 4y2 = 8 ] 2
x2 + 2y2 = 4 or y= ?
enters y = ? , leaves y = ?
4. Finally we find the x-limits of integration
At y = 0
x = 2
Let x = 2sin? dx = 2cos? d?
University of Babylon Lecture: Hussein ALbermany
Collage of Material Engineering Subject: Mathematics
Department of Metallurgy Engineering Stage : 2nd stage
20
Average Value of a Function in Space
The average value of a function F over a region D in space is defined by the formula
Average value of F over D =
EXAMPLE 12: Find the average value of F(x, y, z) = xyz over the cube bounded by the coordinate planes and the planes x = 2, y = 2 and z = 2 in the first octant.
Solution:
? ketch the cube with enough detail to show the limits of integration in the figure
? The volume of the cube is (2).(2).( 2) = 8
? The value of the integral of F over the cube is