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Triple integral volume problem, volume between 2 paraboloids

  • #1

Homework Statement



Find the volume of the solid region E bounded by the paraboloids z = 1+x^2+y^2 and
z = 4 - 2x^2 - 11y^2





The Attempt at a Solution


i set up a triple integral using Cartesian coordinates but was unable to solve it because the limits of integration where very hard to integrate, i don't think you can use cylindrical coordinates because the intersection is an eclipse: x^2 + 4y^2 = 1. PLease somebody show me a good method to do this question as it was in a past exam and might come up again.

Please help me. thanks.
 

Answers and Replies

  • #2
1,796
53
Know what, I didn't do it all but doesn't look like the regular way would be too hard.
[tex]4\mathop\iint\limits_{\text{1/4 my ellipse}} \left(z_1(x,y)-z_2(x,y)\right) dydx[/tex]

and since you have the equation of the ellipse:

[tex]x^2+4y^2=1[/tex]

not hard to get the upper limit on y right? Then x just goes from one side to the other.

What happens when you do that?
 
  • #3
Know what, I didn't do it all but doesn't look like the regular way would be too hard.
[tex]4\mathop\iint\limits_{\text{1/4 my ellipse}} \left(z_1(x,y)-z_2(x,y)\right) dydx[/tex]

and since you have the equation of the ellipse:

[tex]x^2+4y^2=1[/tex]

not hard to get the upper limit on y right? Then x just goes from one side to the other.

What happens when you do that?
is the limits for y: 0<= y < = (√(1-x^2))/2 ?? and then for x it is 0<=x<=1
 
  • #4
1,796
53
Yes.
 
  • #5
thanks jackmell i really like how you split it up to a quarter and multiplied by 4 to take advantage of the symmetry. NOw the integral is not as complicated as the one i initially had when i went from -(√(1-x^2))/2 <= y <= (√(1-x^2))/2
 

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