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!

Double (volume) integral

  1. Oct 5, 2008 #1
    1. The problem statement, all variables and given/known data
    - Sketch the region V of 3-space that is bounded above and below by the two surfaces z=Z1(x,y) = 0 and z=Z2(x,y)=1+x^2+y^2 and where the domain of these functions is the region R in the xy plane enclosed by the four lines y=x, y=-x, y=2+x and y=2-x

    -Calculate the volume using a double integral namely

    http://img145.imageshack.us/img145/2705/q3tv3.png [Broken]
    3. The attempt at a solution

    For the first part I have firstly drawn the four lines in the xy plane
    http://img239.imageshack.us/img239/7389/q3ct1.png [Broken]

    Obviously I make it 3d by adding the top (i.e. 1+x^2+y^2 for -1 ≤ x ≤1 and 0 ≤ y ≤ 2)

    My question is regarding the volume integral. It is difficult to nominate the y bounds due to the nature of the 4 lines. I am thinking I should split the area into two sections, bisected by the y axis such that the volume can be calculated by:

    http://img73.imageshack.us/img73/8666/q3ao8.png [Broken]

    Am I on the right track?
    Last edited by a moderator: May 3, 2017
  2. jcsd
  3. Oct 6, 2008 #2


    User Avatar
    Homework Helper
    Gold Member

    Looks like a good plan to me!:approve:
  4. Oct 6, 2008 #3
    Thanks for the quick reply

    hmmm, noticed that I need to reverse the order of integration (noting also I must change the terminals) though as before I was ending up with x's after the second integration. That is, I need to perform integration with respect to y then with respect to x such that

    http://img98.imageshack.us/img98/5985/q3ao8iu3.png [Broken]
    Last edited by a moderator: May 3, 2017
  5. Oct 6, 2008 #4


    User Avatar
    Homework Helper
    Gold Member

    Yes, I thought your original order of integration was a typo.
  6. Oct 6, 2008 #5
    also noticed my terminals are wrong!

    for the first part with respect to y I should be integrating from x to -x+2 and for the second part I should be integrating from -x to x+2.

    After fixing it up I got a volume of 14/3 units cubed. Is there any way to check this answer?
    Last edited: Oct 6, 2008
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook