1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Volume of a region

  1. Apr 13, 2015 #1
    1. The problem statement, all variables and given/known data

    2. Relevant equations


    3. The attempt at a solution
    1) Whenever I need to find the volume between two surfaces, the integrand is simply the difference (subtraction) of the two equations? In the solution guide above, it is clear that they subtracted the two equations for z.

    2) After transforming the double integral into polar coordinates, how did the solutions guide figure out the limits of integration? The object being integrated is a paraboloid limited by z = 4. Why then do the limits go from 0 to 2pi? Where do the limits for R (the inner integral) come from?
  2. jcsd
  3. Apr 13, 2015 #2


    User Avatar
    Science Advisor

    You shouldn't have to ask! You can take a small "delta x- delta y" rectangle in the xy-plane and then the height of the rectangular solid is the z distance between the bottom and the top- that is, the difference between "the two equations". The volume is z delta x delta y which, in the limit becomes the integral of the z difference dx dy.

    In your uv- coordinates the two bounding surfaces are z= 4 and z= u^2+ v^2. They intersect at u^2+ v^2= 4. You should be able to recognize that as a circle in the uv- plane with center at (0, 0) and radius 2. To cover that circle, take r from 0 to 2 and [itex]\theta[/itex] from 0 to [itex]2\pi[/itex].
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted