Almost finished with Calculus for ever

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SUMMARY

The discussion focuses on calculating the area of a region defined by the plane equation x + 2y + 2z = 5 and the cylinder boundaries x = y² and x = 2 - y. Participants emphasize the importance of visualizing the graphs of the equations to identify their intersection points. The integration limits for the area calculation are critical, and various methods for integrating the differential area of the specified plane are suggested. The conversation encourages users to explore different integration techniques based on the defined boundaries.

PREREQUISITES
  • Understanding of multivariable calculus concepts, specifically integration over regions.
  • Familiarity with the equations of planes and cylinders in three-dimensional space.
  • Ability to sketch and analyze graphs of quadratic and linear equations.
  • Knowledge of differential area elements in calculus.
NEXT STEPS
  • Study methods for visualizing intersections of curves and surfaces in three dimensions.
  • Learn about setting up double integrals for area calculations in multivariable calculus.
  • Explore techniques for integrating over non-rectangular regions.
  • Investigate the application of Jacobians in changing variables for integration.
USEFUL FOR

Students and educators in calculus, mathematicians focusing on multivariable integration, and anyone interested in geometric interpretations of mathematical equations.

Snakeonia
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i hope anyway lol here's my problem

Find area of Region cut from plane x + 2y + 2z = 5
by the cylinder whose walls are x = y^2 and x = 2 - y

can someone point me in the right direction
 
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Start by drawing a picture- what do the graphs of x= y2 and x= 2- y look like? Where do they intersect? If you were integrating over that region, what would the limits of integration be? And, of course, you want to integrate the "differential of area" for the plane x+ 2y+ 2z= 5. Since that is a plane, there are several different ways of computing that. What have you tried?
 

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