Equation of curves (intersection)

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SUMMARY

The discussion focuses on finding the Cartesian equations of the curves formed by the intersection of the paraboloid defined by the equation z = x^2 + y^2 - 4 with the coordinate planes: xy, xz, and yz. The solid region S is bounded by the paraboloid for z ≤ 0, with constraints 0 ≤ x ≤ √(4 - y^2) and 0 ≤ y ≤ 2. The primary goal is to derive the intersection equations by substituting the respective plane equations into the paraboloid equation.

PREREQUISITES
  • Understanding of Cartesian coordinates and equations
  • Knowledge of paraboloids and their properties
  • Familiarity with intersection of surfaces in three-dimensional space
  • Basic skills in algebraic manipulation and solving equations
NEXT STEPS
  • Study the properties of paraboloids and their intersections with planes
  • Learn how to derive intersection curves from three-dimensional surfaces
  • Practice solving similar problems involving solid regions and bounded surfaces
  • Explore graphical representations of surfaces and their intersections
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Students studying calculus, geometry, or physics, particularly those focusing on three-dimensional shapes and their intersections, as well as educators seeking to enhance their teaching of these concepts.

madachi
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Homework Statement



Let [itex]S[/itex] be the solid region bounded by the paraboloid [itex]z = x^2 + y^2 - 4[/itex] for [itex]z \le 0[/itex], [itex]0 \le x \le \sqrt{4 - y^2}[/itex] and [itex]0 \le y \le 2[/itex]. Find the Cartesian equations of the curves if the surface S intersects the [itex]xy,xz[/itex], and [itex]yz[/itex] planes.


The Attempt at a Solution



Is the question asking us to find the equation of the intersection of [itex]z = x^2 + y^2 - 4[/itex] and [itex]z=z[/itex], [itex]z=x[/itex], and [itex]z=y[/itex] respectively?

As I am confused with this question and not really sure how to start, so there is no attempt yet but I had some thought about this question. Could you tell me some hints to start the problem?

Thanks!
 
Last edited:
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i would take the xy plane to be where z = 0
 

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