Calculating Volume Using Double Integrals: Finding the Boundaries and Limits

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SUMMARY

The discussion focuses on calculating the volume of solids defined by two surfaces, S1 (z = 50 - x²) and S2 (z = 9y² + 16), using double integrals. The volumes V1, V2, and V3 are bounded by specific planes and a vertical cylinder, with V1 defined by the planes x = ±1 and y = ±1, V2 by the cylinder x² + y² = 1, and V3 by the same surfaces without additional boundaries. The user sought guidance on solving these problems effectively, emphasizing the importance of understanding the steps involved in double integration.

PREREQUISITES
  • Understanding of double integrals in calculus
  • Familiarity with surface equations and their graphical representations
  • Knowledge of volume calculation techniques using integration
  • Experience with cylindrical coordinates for volume problems
NEXT STEPS
  • Study the method of double integrals for volume calculation
  • Learn how to set up and evaluate double integrals over bounded regions
  • Explore the application of cylindrical coordinates in integration
  • Practice problems involving surfaces and their intersections
USEFUL FOR

Students preparing for calculus exams, particularly those focusing on volume calculations using double integrals, as well as educators teaching integration techniques in mathematics.

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Homework Statement




given two surfaces S1={(x,y,z)|z=50-X^2} S2={(x,y,z)|z=9y^2+16} find the volume

1.V1 bounded above by S1 and below by S2 and on the sides by the vertical planes X=1 X=-1 Y=1 Y=-1

2 the solid V2 bounded above by S1 and below by S2 and on the sides by the vertical cylinder X^2+y^2=1

3.the solid V3 which is bounded above by the surface S1 below by S2

Homework Equations





The Attempt at a Solution



hey Please give me a help because i don't have much time !exam is today and want to understand these kind of questions !it will be nice if u can give me the steps to solve this kind of problem rather that giving answers to this questions!
 
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The exam is today and you have no idea how to even start these problems? (1), at least, is almost trivial. I would expect a problem like that to be in an earlier chapter from (2) and (3). Please try. You learn mathematics by doing mathematics, not by watching others do it.
 
ok thanks !i found the way !
 

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