section 16 3 triple integrals
Download
Skip this Video
Download Presentation
Section 16.3 Triple Integrals

Loading in 2 Seconds...

play fullscreen
1 / 12

Section 16.3 Triple Integrals - PowerPoint PPT Presentation


  • 62 Views
  • Uploaded on

Section 16.3 Triple Integrals. A continuous function of 3 variables can be integrated over a solid region, W , in 3-space just as a function of two variables can be integrated over a flat region in 2-space We can create a Riemann sum for the region W

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'Section 16.3 Triple Integrals' - aloha


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
slide2
A continuous function of 3 variables can be integrated over a solid region, W, in 3-space just as a function of two variables can be integrated over a flat region in 2-space
  • We can create a Riemann sum for the region W
    • This involves breaking up the 3D space into small cubes
    • Then summing up the functions value in each of these cubes
slide3
If
  • then
  • In this case we have a rectangular shaped box region that we are integrating over
slide4
We can compute this with an iterated integral
    • In this case we will have a triple integral
  • Notice that we have 6 orders of integration possible for the above iterated integral
  • Let’s take a look at some examples
example
Example
  • Find the triple integral

W is the rectangular box with corners at (0,0,0), (a,0,0), (0,b,0), and (0,0,c)

example1
Example
  • Sketch the region of integration
example2
Example
  • Find limits for the integral

where W is the region shown

slide8
z

z

y

x

y

x

This is a quarter sphere of radius 4

z

z

x

x

y

y

triple integrals can be used to calculate volume
Triple Integrals can be used to calculate volume
  • Find the volume of the region bounded by z = x + y, z = 10, and the planes x = 0, y = 0
  • Similar to how we can use double integrals to calculate the area of a region, we can use triple integrals to calculate volume
    • We will set f(x,y,z) = 1
example3
Example
  • Find the volume of the pyramid with base in the plane z = -6 and sides formed by the three planes y = 0 and y – x = 4 and 2x + y + z =4.
example4
Example
  • Calculate the volume of the figure bound by the following curves
some notes on triple integrals
Some notes on triple integrals
  • Since triple integrals can be used to calculate volume, they can be used to calculate total mass (recall Mass = Volume * density) and center of mass
  • When setting up a triple integral, note that
    • The outside integral limits must be constants
    • The middle integral limits can involve only one variable
    • The inside integral limits can involve two variables
ad