Volume under the elliptic paraboloid

  • Thread starter Thread starter Damascus Road
  • Start date Start date
  • Tags Tags
    Paraboloid Volume
Click For Summary
SUMMARY

The volume under the elliptic paraboloid defined by the equation z = x² + 4y² and above the rectangle R = [-1, 1] × [-4, 4] can be calculated using a double integral approach. The integration limits for z are determined by the function itself, allowing for the evaluation of the double integral directly over the specified rectangle. The discussion indicates that the user initially considered a triple integral but concluded that a double integral suffices for this calculation.

PREREQUISITES
  • Understanding of double integrals in multivariable calculus
  • Familiarity with the concept of elliptic paraboloids
  • Knowledge of integration limits in rectangular coordinates
  • Basic skills in evaluating integrals
NEXT STEPS
  • Study the method for calculating volumes under surfaces using double integrals
  • Explore the properties and applications of elliptic paraboloids in geometry
  • Learn about setting up and evaluating triple integrals for more complex volumes
  • Review examples of integration over rectangular regions in multivariable calculus
USEFUL FOR

Students and educators in mathematics, particularly those focusing on calculus and multivariable functions, as well as professionals involved in mathematical modeling and analysis.

Damascus Road
Messages
117
Reaction score
0
Greetings again,

this is another I'm stuck on...

Calculate the volume under the elliptic paraboloid z = x^{2} + 4y^{2} and over the rectangle R = [-1, 1] \times [-4, 4].

I'm not sure how to find the limits of z for the triple integral. Can I somehow integrate the function z, and only do a double integral? Perhaps going from -1, 0 and -4, 0 and then double it?
 
Physics news on Phys.org
I believe I figured it out now, never mind.
 

Similar threads

Calculating volume between two paraboloids
  • · Replies 3 ·
Replies
3
Views
3K
The volume of a "spherical cap" using triple integrals
  • · Replies 9 ·
Replies
9
Views
2K
Finding Volume of Solid Bounded by Paraboloid and Planes
  • · Replies 2 ·
Replies
2
Views
2K
The volume of water gathered by a tilted paraboloid antenna
  • · Replies 10 ·
Replies
10
Views
3K
Triple Integral of y^2z^2 over a Paraboloid: Polar Coordinates Method
  • · Replies 6 ·
Replies
6
Views
2K
Volume bounded by two surfaces, what am I missing?
  • · Replies 6 ·
Replies
6
Views
2K
Evaluate the triple integral paraboloid
  • · Replies 3 ·
Replies
3
Views
4K
Volume integral in cylindrical coordinates
  • · Replies 9 ·
Replies
9
Views
2K
Calculating Volume of a Region: Integration in Polar Coordinates
  • · Replies 1 ·
Replies
1
Views
2K
Find volume of solid elliptic paraboloid using polar coordinates
  • · Replies 2 ·
Replies
2
Views
4K