Surface Area Problem: Find Area of Paraboloid Cut by Plane y=25

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SUMMARY

The discussion focuses on calculating the surface area of the paraboloid defined by the equation y = x² + z², truncated by the plane y = 25. The user correctly identifies that this is equivalent to analyzing the paraboloid z = x² + y² cut by the plane z = 25. The surface area can be computed using the integral ∫∫√((dz/dx)² + (dz/dy)² + 1) and is simplified in polar coordinates to ∫∫(1 + 4r²) with r ranging from 0 to 5 and θ from 0 to 2π. The user seeks clarification on projecting the surface onto the x-z plane.

PREREQUISITES
  • Understanding of surface area integrals in multivariable calculus
  • Familiarity with polar coordinates and their application in integration
  • Knowledge of partial derivatives and their role in surface area calculations
  • Concept of projecting surfaces onto different coordinate planes
NEXT STEPS
  • Study the application of double integrals in calculating surface areas
  • Learn about polar coordinate transformations in multivariable calculus
  • Explore the concept of surface projections in three-dimensional geometry
  • Investigate the use of Jacobians in changing variables during integration
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Students and educators in mathematics, particularly those studying multivariable calculus, as well as anyone interested in geometric applications of calculus in three-dimensional space.

bodensee9
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Homework Statement


I am wondering if someone could help me with the following? I am supposed to find the area of the finite part of the paraboloid y = x^2+z^2 that's cut off by the plane y = 25. Now, wouldn't this be the same as the paraboloid z = x^2+y^2 that's cut off by the plane z = 25?

So, if that's right, then the surface area is ∫∫√((dz/dx)^2+(dz/dy)^2+1).
This seems easier to do in polar coordinates, so we basically have the following:

∫∫(1+4r^2)? And, r would be from 0 to 5, and 0≤θ≤2*pi?

Am I doing something wrong here? Or, how do I "project" the surface unto the x-z plane? Thanks!
 
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